A Journey Through Tides [1 ed.] 0323908519, 9780323908511

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A JOURNEY THROUGH TIDES

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A JOURNEY THROUGH TIDES

Edited by

MATTIAS GREEN School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom

JOÃO C. DUARTE Geology Department and Instituto Dom Luiz (IDL), Faculty of Sciences of the University of Lisbon, Lisbon, Portugal

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-90851-1 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice G Janco Acquisitions Editor: Louisa Munro Editorial Project Manager: Sara Valentino Production Project Manager: R.Vijay Bharath Cover Designer: Miles Hitchen Typeset by STRAIVE, India

Dedication This book is dedicated to the editors’ wives, Clare and Noemie.

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Contents Contributors Editors biography Preface Acknowledgments

SECTION 1 1.

xiii xv xvii xix

Fundamentals

Tidal science before and after Newton

3

Philip L. Woodworth

2.

1. Introduction 2. Aspects of the tides known since antiquity 3. Investigations of the tides before Newton 4. Isaac Newton's Principia Mathematica 5. Essays for the Academie Royale des Sciences 6. Before and after Newton 7. Conclusions Acknowledgments References

3 4 9 23 24 28 30 32 32

Introducing the oceans

37

Yueng-Djern Lenn, Fialho Nehama, and Alberto Mavume

3.

1. Our blue planet 2. Physical properties of seawater 3. Geography and ocean circulation 4. Key water masses and global distributions 5. Oceanic impact on and sensitivity to Earth's climate Acknowledgments References

37 40 46 53 57 61 61

A brief introduction to tectonics

65

João C. Duarte 1. Tectonics 2. Earth's tectonic cycles References

65 73 78

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4.

Contents

Why is there a tide?

81

Sophie Ward, David Bowers, Mattias Green, and Sophie-Berenice Wilmes 1. Introduction to tides 2. Tidal theories 3. Tides in the real world 4. Tidal energetics and energy losses 5. Chapter summary References

SECTION 2 5.

81 90 97 107 110 112

A tidal journey through time

A timeline of Earth's history

117

João C. Duarte 1. Geological time 2. Chrono-stratigraphy 3. The geological timescale 4. Main events in Earth's history 5. Final remarks References

6.

Hadean and Archean (4600–2500 Ma)

117 119 121 122 128 129

133

Hannah S. Davies, João C. Duarte, and Mattias Green 1. Introduction 2. Methods 3. Results 4. Discussion References

7.

Proterozoic (2500–541 Ma)

133 135 137 139 139

143

Mattias Green, Christopher Scotese, and Hannah S. Davies 1. Introduction 2. Methods 3. Results 4. Summary Acknowledgments References

143 145 149 152 153 153

Contents

8.

Phanerozoic (541 Ma-present day)

ix

157

Mattias Green, David Hadley-Pryce, and Christopher Scotese 1. Introduction 2. Tectonics 3. “It's life, Jim, but not as we know it” 4. The ups and downs of phanerozoic tides 5. Methods 6. Results 7. Case studies 8. Summary Acknowledgments References

9.

Present day: Tides in a changing climate

157 158 160 161 161 165 172 177 179 179

185

Sophie-Berenice Wilmes, Sophie Ward, and Katsuto Uehara 1. Introduction 2. Climate and sea level through the late Quaternary 3. Modeling the tides during the late Pleistocene and Holocene 4. Tides during the late Pleistocene, Holocene, and into the future 5. Summary References

10. Into the future

185 186 198 200 219 220

231

Hannah S. Davies, João C. Duarte, and Mattias Green 1. Introduction 2. Methods 3. Results 4. Discussion References

SECTION 3

231 233 235 240 243

Consequences of living on a tidal planet

11. Tides at a coast

247

Jennifer M. Brown, Angela Hibbert, Lucy M. Bricheno, Elizabeth Bradshaw, and Amani E. Becker 1. 2. 3. 4.

Introduction Tides at the coast Tidal interactions with other physical processes Transport of matter

247 247 249 256

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Contents

5. Tidal observations at the coast 6. Tidal applications Acknowledgments References

12. Tidal rhythmites: Their contribution to the characterization of tidal dynamics and environments

261 271 276 276

283

Bernadette Tessier 1. Introduction 2. Tidalites and tidal rhythmites: Definition and first description 3. Methodology for tidal rhythmite recognition 4. Environments of deposition of tidal rhythmites 5. Implications of tidal rhythmite recognition and interpretation 6. Conclusion Acknowledgments References

13. Tides: Lifting life in the ocean

283 283 288 289 292 299 299 299

307

Alex J. Poulton 1. The productive ocean 2. The biological carbon pump 3. A nutrient-rich interior ocean 4. A nutrient-limited surface ocean 5. Mixing nutrients up 6. Mixing life down 7. Shining light in the deep 8. Succession and mortality 9. Ecosystem productivity 10. A role for tides, turbulence, and deep production References

14. Tides, earthquakes, and volcanic eruptions

307 310 314 315 318 321 322 323 325 326 327

333

Stephanie Dumont, Susana Custódio, Simona Petrosino, Amanda M. Thomas, and Gianluca Sottili 1. Introduction 2. Data and methods to study the tidal influence on faults and volcanoes 3. Case studies of tidal control on earthquakes and volcanoes 4. How do tides influence seismic and volcanic activity?

333 336 346 357

Contents

5. Summary and future outlook Acknowledgments References

15. Solid Earth tides

xi 359 361 362

365

Harriet C.P. Lau and Michael Schindelegger 1. Introduction 2. Traditional theory and inferences from observations 3. Tides on a complicated Earth 4. Constraining Earth's structure 5. Future tidal study References

16. Atmospheric tides—An Earth system signal

365 367 374 381 382 383

389

Michael Schindelegger, Takatoshi Sakazaki, and Mattias Green 1. Introduction 2. Solar tides 3. Lunar tides 4. Importance of atmospheric tides 5. Beyond Earth's modern atmosphere Acknowledgments References

17. Tidal drag in exoplanet oceans

389 392 398 402 408 411 412

417

Rory Barnes 1. Introduction 2. Water in the cosmos 3. Exoplanet oceans 4. Ocean tides on exoplanets 5. Summary References Index

417 418 421 426 432 435 441

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Contributors Rory Barnes University of Washington, Seattle, WA, United States Amani E. Becker National Oceanography Centre, Liverpool, United Kingdom David Bowers School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Elizabeth Bradshaw National Oceanography Centre, Liverpool, United Kingdom Lucy M. Bricheno National Oceanography Centre, Liverpool, United Kingdom Jennifer M. Brown National Oceanography Centre, Liverpool, United Kingdom Susana Custo´dio Instituto Dom Luiz (IDL), Faculdade de Ci^encias, Universidade de Lisboa, Lisbon, Portugal Hannah S. Davies Helmholtz Center Potsdam, GFZ German Research Center for Geosciences, Potsdam, Germany Joa˜o C. Duarte Geology Department and Instituto Dom Luiz (IDL), Faculty of Sciences of the University of Lisbon, Lisbon, Portugal Stephanie Dumont Instituto Dom Luiz (IDL), Universidade da Beira Interior, Covilha˜, Portugal Mattias Green School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom David Hadley-Pryce School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Angela Hibbert National Oceanography Centre, Liverpool, United Kingdom Harriet C.P. Lau Department of Earth and Planetary Science, University of California Berkeley, Berkeley, CA, United States Yueng-Djern Lenn School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom

xiii

xiv

Contributors

Alberto Mavume Escola Superior de Ci^encias Marinhas e Costeiras, Universidade Eduardo Mondlane, Quelimane, Mozambique Fialho Nehama Escola Superior de Ci^encias Marinhas e Costeiras, Universidade Eduardo Mondlane, Quelimane, Mozambique Simona Petrosino Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Napoli—Osservatorio Vesuviano, Naples, Italy Alex J. Poulton The Lyell Centre for Earth and Marine Sciences, Heriot-Watt University, Edinburgh, United Kingdom Takatoshi Sakazaki Graduate School of Science, Kyoto University, Kyoto, Japan Michael Schindelegger Institute of Geodesy and Geoinformation, University of Bonn, Bonn, Germany Christopher Scotese Earth and Planetary Sciences, Northwestern University, Evanston, IL, United States Gianluca Sottili Dipartimento di Scienze della Terra, Sapienza-Universita` di Roma, Rome, Italy Bernadette Tessier CNRS, Normandie Univ, UNICAEN, UNIROUEN, M2C, Caen, France Amanda M. Thomas Department of Earth Sciences, University of Oregon, Eugene, OR, United States Katsuto Uehara Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka, Japan Sophie Ward School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Sophie-Berenice Wilmes School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Philip L. Woodworth National Oceanography Centre, Liverpool, United Kingdom

Editors biography Mattias Green is a physical oceanographer working on changes in tides and tidally driven processes on a range of scales. After earning degrees in oceanography at G€ oteborg University in Sweden, he moved to Bangor University in 2005. After a stint as a postdoc there, he received an NERC Advanced Fellowship, and later became senior lecturer in physical oceanography at Bangor. He is now professor of physical oceanography at Bangor and works on tides and tidally driven processes and how they change on a range of time scales—from years to eons. He has published over 80 peer-reviewed papers to date and his work has been highlighted by Nature, Science, The Conversation, CNN, and a number of other international outlets. Joa˜o C. Duarte works in tectonics, geodynamics, and marine geology. He is an assistant professor at the University of Lisbon and researcher at IDL, where he coordinates the research group on Continental Margins and the Deep Ocean Frontier. He is also member of the editorial board of Communication Earth & Environment and a corresponding member of the Lisbon Academy of Sciences. Joa˜o has published over 40 papers and has several edited works, which include an Elsevier book entitled Transform Plate Boundaries and Fracture Zones, an Elsevier special volume in the Journal of Geodynamics on “200 years of geodynamic modelling” and an AGU monograph entitled “Plate Boundaries and Natural Hazards.” He was awarded the Discovery Early Career Researcher Award by the Australian Research Council in 2015, and the 2017 Arne Richter Award for Outstanding Early Career Scientists of the European Geosciences Union. Joa˜o is passionate about science communications and he regularly collaborates with science magazines and the media.

xv

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Preface “You work with tides? That must be easy: they come and go twice a day— what else is there to them?” This is a comment we have heard a few times from members of the public, and it shows the view most people have of tides. But there is a bit more to them than the obvious once or twice daily change in sea level. The ocean tides influence a range of Earth system processes: they provide energy that sustains a climate-regulating part of the ocean circulation, they drive ocean primary production in shallow and deep waters, they move sediment and pollutants through the ocean, and they may have set the environment for key evolution events. The tide also slows down Earth’s spin, leading to longer days, which means the Moon must move away from Earth in an effort to conserve angular momentum, so the tide is the key controller of the Earth’s and the Moon’s orbits. And then there are tides that appear not in the ocean but in the solid Earth or in the atmosphere. We think there are several reasons to investigate tides, and in this book, we have collated a series of chapters covering several aspects of tides and their consequences. But tides have not always behaved like they do now. We know that tides today are very large and they must have been smaller for long periods of the Earth’s history. It turns out that the most important controller of tides is tectonics: the location of continents sets the size of ocean basins, and basins of the right size can host very large tides. So, when continents move around in what is known as the supercontinent cycle, the tides respond with a supertidal cycle. Because tides matter for other parts of the Earth system, changes in tides can be important for a range of processes. The editors have worked with linking tides and tidally driven processes to other Earth system processes, particularly geology, for the past decade. In this book, they drew upon their own research experience and that of international experts in their respective disciplines to ensure the story of how tides have changed throughout Earth’s history is available in one volume. But, of course, there is more to it than that: the reader must be eased into the topic, so we start with an introduction to tides and the Earth. Then we tell our story about how tides have evolved over the eons and touch upon why that matters, before diving into the consequences of living on a planet with tides, including discussions of tides outside of the oceans (and Earth).

xvii

xviii

Preface

We have had a great time putting this book together and we have learned a lot from it. It is our hope that the reader will find the topic as engaging as we do and be fascinated and learn a lot from this book as well. Mattias Green School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Joa˜o C. Duarte Faculty of Sciences of the University of Lisbon, Lisbon, Portugal

Acknowledgments A volume like this could not have been produced without the expertise of the chapter authors, and their patience and willingness to contribute is acknowledged and greatly appreciated. Then there is the wider group whose work has inspired, motivated, and pushed us forward: Professor Stephen Balbus (University of Oxford) and Dr. Michael Way (NASA) deserve special mentions, as do the Devonian Tides Group Meetings that really pushed the boundaries of what we can do. Professor Dietmar Muller (University of Sydney) responded to a query about future Earth that led to our collaborations, including this book. There are numerous others who deserve a mention, but rather than risking leaving anyone out, we say a collective thank you to you all. The book came about after a meeting with Louisa Munro at Elsevier, and her faith in us being able to pull this off (despite Mattias being late to the meeting) is very much appreciated. A thank you for her patience and ability to herd cats goes to Sara Valentino, the production editor. Finally, but certainly not the least, we express our everlasting gratitude to our families for their endless support. Here’s to Clare, Jacob, Lucas, Noemie, Lourdes, Joaquim, Manela, Helena, Pedro, and Diogo. Mattias and Joa˜o

xix

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SECTION 1

Fundamentals

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CHAPTER 1

Tidal science before and after Newton Philip L. Woodworth

National Oceanography Centre, Liverpool, United Kingdom

1. Introduction I was asked to write on how ideas on the tides have changed through history. That would have been an interesting challenge had David Cartwright’s excellent book on the history of tidal science not already existed. That book provides a comprehensive overview of investigations into tides from antiquity to the present day (Cartwright, 1999). Rather less well known is his later journal paper which attempted to make up for omissions in the book due to space limitations (Cartwright, 2001). That paper was concerned with findings during the classical era and up to the 13th century. The two publications of Cartwright taken together provide as much detail as most people would require on the history of investigations into tides up to the Middle Ages. Not much of note on tides occurred between the 13th and 16th centuries, although the heliocentric theory of Nikolaus Copernicus (1473–1543), published in the De Revolutionibus Orbium Coelestium (On the Revolutions of the Celestial Spheres) just before his death, was an essential precursor to the Copernican Revolution and so to the work on tides by Kepler, Galileo, and Newton that followed. Therefore, in the present chapter, I have decided to focus on two contrasting periods in later years (i.e., during the 16–17th and 18th centuries) when there was activity, if not progress necessarily, by a small number of researchers on the tides. These two periods are either side of the great leap forward in tidal insight provided by Isaac Newton. In the first period, science without decent physical theories and without the rigor of mathematics was little more than speculation by a well-resourced few. It was made worse by some investigators forgetting, or choosing to ignore, findings from observations on the tides which had been known for centuries. By the second period, science had benefited from the theoretical insight provided by Newton, although his theory was still not accepted universally. A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00002-9

Copyright © 2023 Elsevier Inc. All rights reserved.

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4

A journey through tides

Nevertheless, there was now more mathematical rigor to the work, instead of the earlier plethora of (reasonable or unreasonable) speculation, and there was greater attention to observational data. In fact, one example to be discussed, the use of Bernoulli’s development of a generic tide table leading to practical tables for northern European ports, involved a combination of both theory and measurements. Nevertheless, there were still important aspects of the tides that had been established previously but were still being forgotten (or ignored) by some people. Before discussing these two periods in Sections 3 and 5 (with some mentions of Newton in Section 4), Section 2 provides a summary of the main “bullet points” concerning the tides which would (or should) have been known to European researchers after the 14th century. Of course, the tides were also of interest elsewhere, such as in India with its history of astronomy (Kak, 1996), which saw the construction of the first “wet dock” at Lothal in the Indus Valley in the third millennium BCE (Pannnikar and Srinivasan, 1971; Nigam, 2006). However, the focus of the present paper will be on research in Europe. Section 6 provides a contrast of the before and after Newton periods, while conclusions are presented in Section 7.

2. Aspects of the tides known since antiquity This section summarizes some of the main findings on the tides which had been accumulated up to the 14th century. • Herodotus (c.484–425 BCE) reported in 440 BCE in his Histories that in the Red Sea “there is an ebb and flow of the tide every day” contrasting with the small tides of the Mediterranean (Wright, 1923). • Aristotle (c.350 BCE) famously tried to understand the four times a day reversals of currents through the Strait of Euripus between Boeotia (mainland Greece) and Euboea Island. Although tidal elevations in most of the Mediterranean are only decimetric, tidal currents in straits can be large (e.g., exceeding 2 m/s in the Strait of Messina, the probable source of the Charybidis whirlpool of Greek mythology). However, in this case and as we know now, weather disturbances would have complicated Aristotle’s investigations, as was fully appreciated only recently (Tsimplis, 1997). Aristotle knew that larger tides were to be found in northern Europe than in the Mediterranean. • From the remarkable (some might say incredible) voyage of Pytheas of Marseille (c.350–285 BCE) in about 325 BCE, from the Bay of Biscay, circumnavigating the British Isles and into the North Sea, and possibly as

Tidal science before and after Newton

5

far as Iceland, one learned that the tides of the Atlantic were considerably larger than in the Mediterranean. The tides around the British coast were said to have a height of 120 ft (36 m), a gross overestimate which Huntley (1980) claims was typical of tidal observations until the 17th century. Pytheas also observed that there were two high tides per lunar day and that their amplitude depended on the phases of the Moon (spring tides). These findings were published in his book On the Ocean, now lost but quoted by other authors. At almost the same time (325 BCE), the army of Alexander the Great was surprised by the large tides of the Indian Ocean and was almost destroyed by a tidal bore on the Indus River. • Seleucus of Seleucia (Baghdad) or of Seleukia (Red Sea) (190–150 BCE) was an eminent astronomer and an arguer for a heliocentric system. His original writings are now lost but were reported by Strabo and others. He remarked that the two tides each day in the Erythrean Sea (Arabian Sea) were not equal (diurnal inequality) and that the inequality was largest when the Moon was off the equator. Therefore, the tides obviously had some dependence on the Moon. • Posidonius (135–51 BCE) travelled in about 100 BCE to Gades (Cadiz) on the Atlantic coast of Spain to study the large tides to be found outside the “Temple of Hercules.” He found them to be twice daily “in strict accordance with the motion of the Moon.” In addition, they were “regular” or “irregular” depending on the Moon’s declination (diurnal inequality), with what are now called spring tides separated by neap tides corresponding with New and Full Moons. In these things, he concurred with Seleucus. Based on information from local people, he also concluded (wrongly) that tides are largest at the summer solstice; this implies however that some knowledgeable local person had made an extended set of measurements. His original writings were lost in the fire of the library at Alexandria in 47 BCE but were included in those of Strabo. • Strabo (63 BCE–24 CE) reported in his Geography of 23 CE on many of the previously mentioned findings, and especially on those of Seleucus and Posidonius, for example, that the tides of the Persian Gulf are diurnal and not semidiurnal. He denigrated the reports of Pytheas, calling him “that arch-falsifier,” although Pytheas had been supported earlier by the respected geodesist and mathematician Eratosthenes of Cyrene (276–195 BCE). Cartwright (2001) suggests that Strabo’s sarcasm of Pytheas probably contributed to the vanishing of his book. Strabo also provided what is arguably the first description of earth tides (water level motion in a well due to tidal strain) based on measurements made by

6

•

•

•

•

•

•

A journey through tides

Posidonius at Cadiz; these observations were unexplained until the work of Chaim Pekeris in 1940 (Ekman, 1993). Pliny the Elder (23–79 CE) in his Natural History encyclopaedia also examined findings from Seleucus and Posidonius, concluding that the effect of the Sun’s tides vary through the year resulting in large equinoctial spring tides. He noted the regular time difference between lunar transit and the next high tide and that the maximum tidal range occurs a few days after New Moon (the age of the tide). Harris (1898) mentions that several other Roman writers, including Julius Caesar (100–44 BCE), made the connection between spring tides and Full Moon, with Seneca (3–65 CE) remarking that equinoctial spring tides, when Moon and Sun are in conjunction on the equator, tend to be larger than other spring tides. The Greek astronomer and mathematician Claudius Ptolemy (100–170 CE) attributed the phenomenon of the tides to a virtue or power exerted by the Moon on the waters (Hecht, 2019; Wikipedia, 2022a). The Venerable Bede of Jarrow Abbey (672–735 CE) made many important observations in a section called On the Harmony of the Sea and Moon in his De Temporum Ratione (The Reckoning of Time) of 725 (Wallis, 2004). He noticed that in 12 lunar months of 354 days the sea rises and falls 684 times and not 708, so the tide relates primarily to the Moon and not the Sun. He remarked on the progression of the tide down the east coast of England, flowing from the “boundless northern sea,” and he observed that every location has its own timing relative to the Moon (now known as its establishment or phase lag). Bede was also aware of the ability of the wind to alter both the time and height of high water. His findings on the relationship between the Moon and the tide were demonstrated in beautiful “tidal rota,” which were in effect tide tables (Fig. 1.1) (Edson, 1996; Hughes, 2003). Similar notions to those of Ptolemy were espoused by the Persian astrologer Ab
u Ma’shar (787–886) (Hecht, 2019), while in a book on the Wonders of Creation, the Arabian scientist Zakariya al-Qazwani (1203–83) attempted to explain that the flowing tide is caused by the Sun and Moon heating the waters and making them expand. However, he failed to explain the dominant role of the Moon (Ekman, 1993). Gerald of Wales (1146–1220) observed that the tides had the same or opposite timings at locations on the Irish Sea coasts in Britain and Ireland, depending on the individual locations, each one with a particular

Tidal science before and after Newton

7

Fig. 1.1 A rota showing the relationship between the Moon and the tide. While Bede’s actual text did not refer to diagrams, in some rota “ut Bede docet” (Bede teaches) is printed below them. This rota probably came from the library at Fleury in France in the late ninth century. In the interpretation of Edson (1996), the Earth is in the center, divided into its three continents and surrounded by the winds. The scribe has filled in the names of eight winds, although 12 spaces are provided. The next ring is numbered with the 29 days of the lunar month divided into four parts representing cycles of the tide of 7 or 8 days. The ring beyond labeled “Aqua” may simply be indicating that the tides are the subject of the diagram. The outer ring contains the Moon’s age, from 1 to 30 days (L. xxii is missing). The four circles in the corners are marked to show when the highest tides (malina, days 13 and 28) and lowest tides (ledona, days 5 and 20) occur each month. See also the description of this particular rota in Hughes (2003). (© The British Library Board (MS Harley 3017, f.135r).)

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A journey through tides

relationship to the time of passage of the Moon across the meridian. These suggestions were consistent with Bede’s ideas of tidal progression along coastlines. • This leads to the “St. Albans Tide Table” of John of Wallingford for the “flood at London Bridge”. Lubbock (1837, 1839) refers to the Benedictine monk John of Wallingford as Abbot John (d. 1213), information that was repeated by Harris (1898), Huntley (1980), and Cartwright (1999). However, it is claimed that this was an error in Lubbock’s 19th century sources, resulting in a confusion between the monk John (d. 1245), the actual collector of the manuscript which contained the tide table, and his earlier namesake the Abbot John (d. 1213/4) (Wikipedia, 2022b). Strictly speaking, this was not a tide table based on any observations but assumed the tide to be 3 h 48 min after lunar transit at New and Full Moon, incrementing by 48 min each day. Nevertheless, this demonstrates the unambiguous association now known to occur between the Moon and the tides. Some of the previously mentioned observations on the tides were accompanied by ad hoc theories for their generation. Seleucus ascribed tides both to the Moon and to a whirling motion of the Earth modified by a “pneuma” (breath or wind). Bede suggested a physical mechanism involving the “Moon blowing on water.” Other theories similarly invoked some kind of “breathing,” “heating,” or “pressing of the atmosphere.” For example, Leonardo da Vinci speculated “as man has in him a pool of blood in which the lungs rise and fall in breathing, so the body of the Earth has its ocean tide which likewise rises and falls every six hours, as if the world breathed.” Many of these theories implied a somewhat implausible change in the total volume of water in the ocean through the tidal cycle, rather than the transfer of water from place to place during that cycle. Harris (1898), Deacon (1971), Cartwright (1999, 2001), and Parker (2010) may be consulted for more on tidal ideas in antiquity. In particular, Harris (1898) contains an extensive set of notes of tidal work and knowledge before the time of Newton (Chapter 5, pp. 386–409), Newton to Laplace (Chapter 6, pp. 410–421), and Laplace (Chapter 7, pp. 422–437). Although written more than a century ago, Cartwright (1999, Chapter 1) considered Harris (1898) to have been the most thorough review of early ideas on tides. Another historical review written at almost the same time can be found in Chapter 4 of Darwin (1899). One would like to think that later researchers would have carried away a few basic facts from this earlier body of work, in particular that tides in most

Tidal science before and after Newton

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places are twice daily and more closely associated with the Moon than the Sun. After all, sailors had known since ancient times that there was some connection between the Moon and the tides, and following the evidence is how science tends to progress. However, this was not always the case, as demonstrated by the set of researchers who preceded Newton in our first period. The work of that small number of investigators is covered by only a few pages in Chapter 4 of Cartwright (1999) so they are worth revisiting.

3. Investigations of the tides before Newton Our story can start with Johannes Kepler (1571–1630), one of the key figures in the scientific revolution of the 17th century. Kepler is famous for his three laws of planetary motion which modified and extended the heliocentric theory of Copernicus and which were later shown by Newton to be consistent with his own three laws of motion and the law of universal gravitation. Harris (1898) mentions that Kepler was forming objections to the tidal ideas of Galileo (see later) as early as 1598. In his De Fundamentis Astrologiae Certioribus (On The More Certain Fundamentals of Astrology) of 1601, Kepler noted that “all things swell up with the waxing Moon and subside when she is waning”. In this book, Kepler made what is thought to be the first mention of a 19-year variation in the tides (see Thesis 47 in the translation of Brackenridge and Rossi, 1979). Whether he had in mind the nodal or, more likely, the metonic cycles of the Moon (periods of 18.6 and slightly more than 19 years, respectively) is not clear. Both would have been known since antiquity, but in fact only the former is important for tides. Nowadays, a book on astrology by such a famous astronomer might seem strange. However, at that time astrology and astronomy were treated together. Kepler himself earned a living from reading horoscopes. However, he was not completely convinced by them, maintaining that “If astrologers sometimes do tell the truth, it ought to be attributed to luck” (CDSB, 2008). In Kepler’s last book, a novel called Somnium (Dream), published posthumously in 1634 but actually written in 1608, he speculated in a clear modern-sounding way “the causes of the ocean tides seem to be the bodies of the Sun and Moon attracting the ocean waters by a certain force similar to magnetism. Of course, the body of the Earth likewise attracts its own waters, an attraction which we call ‘gravity’.” A year later, Kepler’s axioms for a “true theory of gravity” in his Astronomia Nova of 1609 included the need for attraction between the Earth and Moon. For this, he looked to a form of magnetic attraction, having been

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A journey through tides

inspired by the publication of William Gilbert in 1600 concerning the magnetic field of the Earth (Ekman, 1993; Fara, 1996; Hecht, 2019; Wikipedia, 2022a). As for the tides, he stated “If the Earth ceased to attract (to itself ) the waters of the sea, they would rise and pour themselves over the body of the Moon.” As a result, he claimed that the tides would be excited “insensibly in enclosed seas, but sensibly where there are broad beds of the ocean.” Kepler’s interpretation later took an apparently backward step when the expression of his astrological views in the Harmonices Mundi (1619) led to him interpreting the tides in terms of the mystical breathing of terrestrial animals and especially the breathing of fish. The CDSB (2008) states that at this time “swept on by his fantasy, Kepler found animistic analogies everywhere”. In addition, Scientific American (1858) reported that Kepler “believed that the earth was a real living animal, that the tides were due to its respirations, and that men and beasts were like insects feeding on its back” but ignored his earlier support for an attraction such as magnetism. However, it does not follow that Kepler had renounced his earlier views of attraction (magnetic or gravitational) (Harris, 1898). There is much more to be said about Kepler. Recent reviews of his life and works can be found in Hecht (2019) and Wikipedia (2022c). It is important to realize how difficult it was for other thinkers to grapple with the idea of attraction or “action at a distance” by some mysterious force such as that proposed by Gilbert or Kepler. For some it almost smacked of the occult (“Occult” is an Aristotelian and early modern term used when distinguishing qualities which are evident to the senses from those which are hidden (Roos, 2001).). In particular, the idea was ridiculed by Galileo who considered it “to be a lamentable piece of mysticism which he read with regret in the writings of so renowned an author as Kepler” (Thomson, 1882). Galileo Galilei (1564–1642) was a champion of the Copernican revolution (Wikipedia, 2022d). He has been called the “father of the scientific method” which, to modern ears, suggests greater attention to reconciling theory with data than was the case with his theory of the tides published in the Discourse on the Tides of 1616 and the Dialogue Concerning the Two Chief World Systems of 1632. Galileo completely dismissed Kepler’s belief that tides were caused by the Moon, a simple fact that had been known since antiquity. Galileo’s theory of the tides has been discussed in the literature far more extensively than have most incorrect theories (e.g., Aiton, 1954, 1963; Burstyn, 1962, 1963; Aiton and Burstyn, 1965; Shea, 1970; Palmieri, 1998). Roos (2001) commented that “there is a virtual academic industry on Galileo and the tides.” The many publications are undoubtedly a reflection of

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Galileo’s otherwise considerable scientific achievements. Galileo persisted with his theory of the tides over many years, even though it is likely that he knew it may be incorrect (Wikipedia, 2022e). The theory has since been categorized, rather kindly, as a “fascinating idea” as a result of the overriding need to provide evidence for the motion of the Earth (Einstein, 1954). Otherwise, it has been described as “Galileo’s big mistake” (Tyson, 2002). In trying to defend the Copernican theory, Galileo suggested that the tides were due to the Earth’s rotation around its axis and its orbital motion around the Sun (Fig. 1.2). The principal causes of them were said to be (1) “… the determinate acceleration and retardation of the parts of the Earth, depending on the combination of two motions, annual and diurnal; …” and (2) “… the proper gravity of the water, which being once moved by the primary cause, then seeks to reduce itself to equilibrium, with repeated reciprocations..”. See Aiton (1954) and Cartwright (1999) for explanation of how such an argument is confused by the mixture of reference frames. Galileo correctly pointed out that large tidal ranges tend to be accompanied by weak tidal currents (and vice versa), characteristic of standing waves, and somehow advanced this observation on the “varieties of tides” as a

Fig. 1.2 Galileo’s theory of the tides was based on his observations of the seiche-like motions of water slopping in a barge when subjected to an acceleration. He attempted to explain the tides by suggesting that the ocean “cavities” (or basins) were similarly subject to such accelerations. EF represents part of the Earth’s orbit around the Sun (period one cycle per year) and its rotation is shown by the arrows (period one cycle per sidereal day). At point A, the annual and diurnal motions are in the same sense, while at point B they are opposite. The absolute speed (relative to the Sun) is therefore greater at A than B, and consequently each part of the Earth’s surface is alternately accelerated and decelerated. (Adapted from Aiton, E.J. 1954. Galileo’s theory of the tides. Ann. Sci., 10, 44–57, https://doi.org/ 10.1080/00033795400200054.)

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A journey through tides

confirmation for nonuniform movement (accelerations) implied by his theory (Harris, 1898). The most obvious problem with Galileo’s theory was that it suggested one (and not two) tides per solar (and not lunar) day. In other words, it suggested the tides to be dominated by what is nowadays called an S1 (solar diurnal) rather than an M2 (lunar semidiurnal) component. The theory therefore failed on two major counts, as had been pointed out to Galileo by Kepler. Galileo hand-waved these problems aside. Cartwright (1999) explained that Galileo was not convinced of the evidence for two tides per (lunar) day at most locations, thereby ignoring the findings of Posidonuius and others, and instead could have been influenced by the tide at Venice having a strong diurnal component. Polli (1952) lists the amplitudes of the M2, S2, K1, and O1 constituents at Ponta della Salute as 23, 13, 16, and 5 cm, giving a Form Factor for the Venice tide of 0.58, which implies a “mixed, mainly semidiurnal” tidal regime (Pugh and Woodworth, 2014). Harris (1898) points out that, to be fair, Galileo had been contemplating a treatise on the theory of the tides, but that the religious persecution of the time would not have enabled him to continue with his scientific work. So we have to wait until much later (1666) when an extended version of Galileo’s theory was proposed by John Wallis (1616–1703), Professor of Geometry at Oxford (Deacon, 1971). Wallis was concerned by the lack of association of the tides to the Moon in Galileo’s theory. He observed correctly that it was the center of gravity of the Earth-Moon system which orbits the Sun. As a consequence, the tides result from the Earth’s rotation combined, not only with the Earth’s motion around the Sun but also with rotation around the center of gravity. Wallis’s suggestion thereby inferred one tide per lunar day, an improvement on Galileo’s one tide per solar day, but still not two tides. Wallis (1666), published in the first volume of the Philosophical Transactions of the Royal Society, has the distinction of being the first paper on tidal theory to appear in a scientific journal. A contemporary of Gilbert was Sir Francis Bacon (1561–1626), Lord High Chancellor of England 1617–21 (SEP, 2021; Wikipedia, 2022f). Bacon claimed that “knowledge is power” and was a deep-thinking individual with a vast range of scientific interests. (There was also a 19th century suggestion called the "Baconian Hypothesis" that Bacon was the real author of Shakespeare’s plays.). He is sometimes called the “father of empiricism” and his ideas published in his influential novel New Atlantis in 1626, for example, are considered as guiding spirits leading to the founding of the Royal Society in 1660 (Fig. 1.3). He believed that knowledge should be

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Fig. 1.3 An etching by Wenceslaus Hollar, after John Evelyn, frontispiece to The History of the Royal-Society of London by Thomas Sprat (1667). On the left is William Brounckner, a mathematician and the first President of the Royal Society; in the center, King Charles II (1630–85); on the right Francis Bacon, 1st Viscount St Alban (1561–1626), philosopher and Lord Chancellor. (©National Portrait Gallery, London.)

based only on careful observations of nature and on inductive reasoning. The Baconian Method, the first formulation of what is now called the scientific method, was introduced in his Novum Organum (New Method) of 1620 and is still of research interest regarding tides and other phenomena (Schwartz, 2017). He is said to have lost his life to pneumonia while researching the effects of freezing on meat preservation.

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A journey through tides

Bacon began his essay of 1623 On the Flux and Reflux of the Sea by recognizing the daily, half-monthly and monthly cycles of the tides, and a halfyearly cycle with greater tides at the equinoxes than at the solstices (Shea, 1970). He suggested that the apparent monthly and annual variability in the tides would be similar everywhere, as is the case. He also noted the progressive wave nature of the tides as they propagated south to north along the eastern coast of the North Atlantic, similar to the observations of Bede along the east coast of England. He made the case for observations elsewhere. Galileo made similar comments on tidal progression although Bacon is believed to have arrived at his own conclusions before news of Galileo’s theory reached him (Aiton, 1954). Aiton (1954) states “This idea that the tides depend on the progressive movement of water and not on any alteration of its physical state is the only positive contribution made by either Bacon or Galileo to the solution of the problem of the tides.” Bacon was one of Galileo’s earliest opponents because of the former’s Ptolemaic Earth-centered, rather than Copernican, perspective. However, in common with Galileo, he seems to have ignored the evidence of tidal cycles and the role of the Moon when it came to devising his theory for the tides. Bacon asserted “I am fully persuaded, and take it almost as an oracle, that this motion (the tides) is of the same kind as the diurnal motion (of the Earth).” As a result, his explanation for the tides involved diurnal motion only, rather than the diurnal and annual combination of Galileo. He observed that all (or most) heavenly bodies moved from east to west every day and the motion was greatest in the heavenly sphere of the fixed stars. Each sphere was considered to affect the motion of the sphere below (i.e., the various planets) with motion decreasing downwards. One eventually reached the level of the atmosphere with its east to west movement of the Trade Winds. Similarly, he considered that ocean currents (however generated) would be a simple westward flow in the absence of continents. The tides occur in this theory as a result of the obstruction of these currents by the continents, where they are reflected and so cause the observed ebb and flow. Because of the westward motion, tides in gulfs or bays which open toward the east on the western sides of ocean basins should have larger tides than elsewhere. In common with Galileo, he had no explanation for the observed two lunar tides per day, claiming that the period was nothing to do with the Moon but was determined by the dimensions of the Atlantic in some kind of resonance akin to the sloshing of water which had led to Galileo’s theory. This sort of idea was not new. The Italian scientist Julius Caesar Scaliger (1484–1558) had suggested some kind of trans-Atlantic

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resonance (or “seiche”) mechanism in 1557 (Ekman, 1993) (See Harris (1898) for more information on Scaliger.). Aiton (1954) provides a discussion of the theories of both Bacon and Galileo and the widespread controversy about them at the time. He points out that while Galileo’s theory of the tides was a failed attempt to prove, once and for all, the validity of the Copernican system, so Bacon’s theory was ultimately a failed attempt to provide conclusive evidence for the Ptolemaic (or Aristotelian) perspective. Some years later (1651), a theory of the tides by William Gilbert (1544–1603, Wikipedia, 2022g) was published posthumously (in Latin) in A New Philosophy of Our Sub-Lunar World. Gilbert had previously proposed that the Earth acts like a large magnet, as published in De Magnete in 1600 (Fara, 1996). He now suggested that the orbits of the planets and the tides were determined by magnetism, and similarly “The Moon produces the movements of the waters and the tides of the sea…” (Ekman, 1993; Hecht, 2019). Bryant (1920) states that Gilbert did not suggest explicitly that there was an attraction between the Moon and water, but more vaguely that “subterranean spirits and humors, rising in sympathy with the Moon, cause the sea also to rise and flow to the shores and up rivers.” Although the lunar, rather than solar, connection was recognized here, the twice daily character of the tides remained unexplained. It is perhaps surprising from a modern perspective to find magnetism, rather than gravity, discussed so much in the context of history of the tides, and to find that Gilbert and then Kepler, among others, had been inspired to propose magnetism as a mechanism for them. However, Fara (1996) explains how the De Magnete of Gilbert was adopted widely as a “magnetic philosophy” that was a central part of 17th century thinking. In addition, Athanasius Kircher (discussed later) was an expert on many philosophical (and apparently magical trickery) aspects of magnetism including a magnetic map of the world (Glassie, 2012; Udı´as, 2020). Newton’s writings included only passing references to magnetism, and yet he was interested enough to own a magnetic signet ring mounted with a powerful chip of lodestone. On a more practical level, by the 18th century we find William Hutchinson, the Liverpool dockmaster, making the case for better magnets in compasses for negotiating the tides (Hutchinson, 1777). It is interesting that, after all this body of work and only a couple of decades before Newton’s Principia was published, respected (in some places) investigators were still coming up with what are now seen to be absurd ideas for the tides. In his book, Cartwright (1999) remarks that it would be “unnecessary [for him] to enlarge on some quite unscientific theories of

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A journey through tides

the tides.”. However, to omit them completely would present a perspective of investigation at that time through a filter of modern insight. Therefore, it seems worthwhile to mention a couple of them here who had a following at the time. Anthanasius Kircher (1602–80) has been described as either “a master of a hundred arts” in his own opinion, or “more of a charlatan than a scholar” in that of Rene Descartes (Brauen, 1982; Findlen, 2004; Glassie, 2012; Wikipedia, 2022h). Either way, he was a fascinating and influential character with interests in many things (especially geology as mentioned later), extremely well-read and a prolific writer with over 30 books, making use of an enormous amount of scientific evidence (real or manufactured) sent to him in Rome by other Jesuits around the world. In line with the religious doctrine of the time, he opposed the Copernican heliocentric proposition and its assumption in the astronomical work of Gilbert and Kepler. He considered their scientific fallacies “pernicious to the Christian Republic and dangerous to the faith of the church” (Baldwin, 1985). Nevertheless, he communicated with a large number of the most important scientists of the mid-17th century via what was called the Republic of Letters (Wikipedia, 2022i). His name is largely forgotten today, probably because, it has to be said, most of his ideas were ridiculous. In the Mundus Subterraneus (Underground World) of 1665, Kircher covered a vast range of Renaissance science and pseudoscience, seeking rational causes for various phenomena through an understanding of natural laws derived from observations rather than miraculous explanations (Wikipedia, 2022j). This lavishly illustrated publication can be inspected at Internet Archive (2022). The mythical whirlpool of Charybdis in the Strait of Messina near the Scylla rock in Calabria, first mentioned by Homer, is discussed at the end of Book 2 (of 12) in terms of winds driving water through an underground channel linking the two sides of Sicily in which they are heated by Mount Etna. Book 3 of 12 is concerned with wider aspects of hydrography. Section 1 discusses general properties of the ocean including its general east to west motion. Tides are covered in Section 2 in which it is clear that Kircher appreciated the basic astronomy of the Moon returning to its apparent position after about 25 h and the combined roles of Moon and Sun in the cycle of New, Quarter, and Full Moons. He knew that the tides had a diurnal and monthly character to them (from which we understand semidiurnal and semimonthly), and he was aware of the large tides outside the Mediterranean such as those experienced by Alexander the Great. He suggested that tides were caused by the effect of the Moon

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on the light of the Sun. The pure light of the Sun would be infected with a “nitrous quality” as it is reflected off the Moon and, passing to the Earth, causes turbulence and a rise in the level of the sea. As a result, the “nitrous effluvia of the Moon” causes water to be pushed and pulled through a global network of “hidden and occult passages” (a main topic of the Mundus Subterraneus discussed at length in the Pyrographicus of Book 4). Section 2 of Book 3 also contains descriptions of different tides at several places including the high tides of London, which he describes (quite reasonably) as due to the tides of Atlantic restricted by passage up the English Channel. It is believed that Kircher probably got his information about the London tides from Sir Robert Southwell (a diplomat, later to be President of the Royal Society) or an earlier English visitor to Rome such as the diarist John Evelyn (1620 1706), another of the founders of the Royal Society (Brauen, 1982; Reilly, 1974). In addition, he refers to the tidal vortex (maelstrom) off the north coast of Norway, located adjacent to another supposed underground passage beneath Scandinavia connecting the Atlantic with the Gulf of Bothnia. He maintained that proof of all this could be demonstrated by observing the “nitrous quality” of the Moon in a small benchtop experiment involving the Moon shining on a basin of sal ammoniac (ammonium chloride). He claimed that an infusion of that volatile salt “placed obliquely to receive the Influence of the Moon ... did Increase and Decrease as it held of an equal Correspondence, by an uninterrupted Chain of Atoms, with the Flowings and Ebbings of the Marine waters.” Furthermore, the effect would be stronger on a moonlit night when the two luminaries (Sun and Moon) were in conjunction or opposition. Roos (2001) suggests that behind his idea might have been the fact that ammonium chloride is hygroscopic. However, the experiment was tried at the Royal Society by Henry Oldenburg (its first Secretary and the first editor of the Philosophical Transactions) and Robert Boyle (one of the founders of modern chemistry), with a visit by Sir Robert Moray (one of the founders of the Society) to see how successful it was (Reilly, 1974; Glassie, 2012; Roos, 2001). Moray had dissolved an ounce of “Bay Salt” and another of niter (saltpeter) in two-and-a-half pints of water and, after staring at it for half an hour, was rewarded with only a few bubbles. Boyle then had his assistant repeat the experiment for two full nights, again a failure. Moray told Oldenburg that he should not bother to communicate such negative experimental results to the Philosophical Transactions “knowing your moments may be better employed,” while Oldenburg concluded that, this first of experiment of Kircher having been a failure, it was likely that all the others in Mundus Subterraneus would be also.

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A journey through tides

A similar-sounding idea was proposed by the English poet and writer Thomas Philipot (d. 1682). His idea can be considered as independent of Kircher’s, and Philipot would probably have been unaware of the Royal Society experiments. He produced his 1673 essay on a chemical theory of the tides, partly as a (justifiable) criticism of Galileo and Kepler and most of the other theories that preceded Newton. The essay included a review of the many competing ideas at that time. He proposed that the “flux of the tide” (its rise) was due to volatile salts “armoniack salt or spirit, that is wrap’d up in the Bowels of the Sea” that were released by the “Impressions of the Sun and Moon.” For the “reflux of the tide,” (its fall) he invoked the action of the “spring of the air,” which was Boyle’s term for air pressure. One should read Roos (2001) for an explanation of the justifications for his theory in the context of the time. His arguments can be considered as a contribution to the then general interest in the chemistry of salts involving Boyle and others, and, as Philipot himself remarked, his theory of the tides was no less absurd than the breathing animal of Kepler. But to return to Kircher, one aspect of tides for which he does deserve credit is his suggestion of the use of a float and stilling well for tidal measurements, a simple technology that remains in use at many locations around the world (Woodworth, 2022). A drawing can be found in Book 3 of the Mundus Subterraneus (Fig. 1.4). The same suggestion was made at almost the same time by Sir Robert Moray in a paper that was also published in the first volume of the Philosophical Transactions (Moray, 1666). Moray is usually given the credit for the idea, but the two suggestions may not have been a coincidence. Moray is known to have read Kircher’s 1641 book Magnes Sive de Arte Magnetica (The Lodestone, or the Magnetic Art), while a prisoner of the Duke of Bavaria in 1643–45. Glassie (2012) states that this began a set of correspondence between Moray and Kircher which lasted decades. For example, Moray’s observations of the tides in the Hebrides, published in the same first volume of the Philosophical Transactions (Moray, 1665), is mentioned in Book 3 of the Mundus Subterraneus. Therefore, it is quite possible that they corresponded about the stilling well idea. Lalande mentions Moray’s tide gauge and the instructions for its use (Lalande, 1781). Lalande also mentions a similar instrument which had been described in an Italian journal in 1675. A summary and diagram derived from that report can be found in the Journal des Sc¸avans of 22 April 1675 (page 118, https://gallica.bnf.fr/ark:/12148/bpt6k56526h/f107.item). This was the earliest academic journal in Europe, starting in January 1665 shortly before the Philosophical Transactions in March.

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Fig. 1.4 A diagram of a float and stilling well tide gauge on page 157 of Book 3 of the Mundus Subterraneous by Athanasius Kircher (Kircher, 1665). (Image courtesy of the Herzog August Bibliothek, Wolfenb€ uttel, Germany.)

Moray (1665) was the first paper on tides to appear in a scientific journal (Fig. 1.5), while Wallis (1666) could be said to be the first paper on tidal theory. Moray (1665) pointed to tidal currents between islands in the Hebrides, as observed by himself and local fishermen, being diurnal in an otherwise semidiurnal tidal regime, an aspect which was not fully understood until recently (see Chapter 13 of Cartwright, 1999). Moray was apparently not pleased that Kircher had printed the contents of a letter informing him of

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A journey through tides

Fig. 1.5 The first page of Moray (1665), the first paper on tides in a scientific journal.

the Hebrides tides. Oldenburg wrote to Moray that he was disappointed that Kircher had been unable to explain their strange character and probably considered that the measurements were defective (Reilly, 1974). If there had been a prize for best theory of the tides at this time, similar to that discussed in Section 5, then the possible winner would have been Rene Descartes (1596–1650), often called “the father of modern philosophy” (Wikipedia, 2022k). His theory of 1644 was the only one to suggest two tides per lunar day, based on a decent attempt at a physical theory of attraction. Aiton (1954, 1955a) and Cartwright (1999) explain how the

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heliocentric theory accepted by Descartes had the planets pulled along in their orbits by a vortex of a rotating ether-like medium generated in turn by a rotating Sun. Similarly, the Earth’s rotation inputs its own local vortex into the ether which carries the Moon along in its monthly orbit. The tides were due entirely to the Moon and occurred through the center of the vortex being displaced slightly from the center of the Earth (a suggestion that was over 20 years in advance of Wallis’s concept of an Earth-Moon center of gravity displaced from the center of the Earth). Rather than explaining the spring-neap cycle as a combination of lunar and solar attraction, Descartes invoked what was then a well-known but small solar perturbation in the Moon’s orbit called “variation.” This perturbation is now known to lead to a minor tidal constituent called μ2, much too small to account for the observed spring-neap cycle. One problem with the theory was that it suggested the passage of the Moon to be accompanied by a low, rather than high water. However, given the known lags in the real tide behind the Moon, this was not necessarily seen as a problem. Newton later demonstrated the theory of Descartes to be incorrect, being inconsistent with Kepler’s third law of planetary motion, and with the fact that any planet carried along by a vortex would have to have the same density and motion as the vortex medium itself. Nevertheless, unlike the hand-waving of some other previous investigators, this was enough of a theory to lend itself to ongoing discussion and development. It survived even after Newton’s death, being the basis of the theory of Antoine Cavalleri in the 17th century (Section 5), its “action by contact” approach being more palatable for some than the “action at a distance” of Newton’s theories. Some of the previously mentioned investigators realized that if progress was to be made then more measurements of the tides were required. Several of the more important of these associated with the Royal Society in London and by the Academie Royale des Sciences in Paris are described in Chapter 6 of Cartwright (1999). For example, Wallis had failed to see why tides should be larger at the equinoxes, a feature of the tides which seems obvious now. Instead, he persisted with reports of them being larger in February and November. This controversy following his 1666 publication led to a call for more measurements of both high and low water heights through the year at ports as close to the open sea as possible (Deacon, 1971). A number of other reports had been inconclusive, largely because of the difficulty of separating tides from the effects of winds and river runoff in short and imprecise sets of measurements. In particular, the publication of Joshua Childrey (1670) rejected Wallis’s claim of tides being larger in February and

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A journey through tides

November, suggesting also that those observations had been the effect of winds rather than tides. He stated that English seamen as a rule believed the largest tides to occur at the equinoxes. In addition, he was the first to observe that high tides also tend to occur when the Moon is close to perigee, in addition to the spring-neap periodicity, resulting in “perigean spring tides.” The first systematic observations of the tides for scientific purposes in the UK, preceding those associated with the Royal Society, were probably those of Jeremiah Horrocks (1618 41) at Toxteth near Liverpool. His tidal measurements are thought to have spanned several weeks in 1640, with the hope of collecting a much longer record. This was prevented by his death in 1641, and his tidal records did not survive the Civil War. Horrocks is most well known for his prediction and observation of the Transit of Venus in 1639. The insight of Horrocks into the orbits of comets and planets and the Moon provided a bridge between the work of Kepler and Newton and undoubtedly contributed to Newton’s thinking in the Principia. Cartwright (1999) refers to some of these individuals (but not Horrocks) as “early amateur observers,” although their observations were in retrospect as important as those of the distinguished scientists of the day. The stir caused at the Royal Society by Wallis’s publication led the Society to charge William Brouncker (its first President) and Moray with organizing a program for measurements at as many locations as possible, such as in the Thames and Bristol Channel where tides are large. That measurement campaign itself never happened. Nevertheless, the standards for measurements which Moray laid down and his suggestions for the use of stilling wells (see previously mentioned) laid the groundwork for future measurements. Reidy (2008) provides a readable account of the controversies at the Royal Society at this time as a result of Wallis’s publication. As for tidal prediction, Henry Philips made a modification to prediction of the time of high waters at London through the spring-neap cycle by introducing (as we would explain now) an addition to the familiar 48 min increment per day of a cosine term with period of 15 days and amplitude 45 min. John Flamstead, the first Astronomer Royal, made use of measurements of the times of high water at Tower-Wharfe in late 1661 and Tower-Wharfe and Greenwich in summer 1682, together with his astronomical insight into the orbits of the Moon and Sun, to produce a tide table for the times of London tides for 1683–88 and, by means of simple adjustments, (somewhat imprecise) times of tides elsewhere.

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4. Isaac Newton’s Principia Mathematica As we have seen, up until this point there had been little constraint on advancing theories on the tides that conflicted with well-established evidence. Any theories tended to be descriptive and lack mathematical rigor. In fact, as Glassie (2012) points out, although Europe had many so-called professors of Mathematics (Kircher was one), mathematicians had traditionally been viewed with condescension by natural philosophers and theologians. In their opinion, mathematics could be used to measure and describe and had some practical applications, but it had nothing to say of the causes or nature of things. All this was to change, as demonstrated by the Restoration in England leading to the founding of the Royal Society in 1660 as a “College for the Promoting of Physico-Mathematical Experimental Learning.” Attention to scientific evidence and use of the power of mathematics were the two keys to the triumph of Isaac Newton’s Philosophiæ Naturalis Principia Mathematica published in 1687. As Cartwright (1999) remarked, scientific measurements would provide the foundation on which theoretical ideas would be built. Newton stated later “Instead of the conjectures and probabilities that are being blazoned about everywhere, we shall finally achieve a natural science supported by the greatest evidence.” Harper (2011) describes in detail Newton’s “experimental philosophy” or what would now be called his “scientific method” as applied to his arguments for universal gravity. Central to that method is the need to test hypotheses using observations of any consequences which flow from them. Mathematics had been essential for Newton reaching the conclusions in the Principia, having been obtained by means of his “infinitesimal calculus,” although for its publication Newton had reworked his arguments in the more widely understood language of geometry. As a result, we arrive at Books One and Three of Newton’s Principia in which he explained in a few pages, and in a supplement called The System of the World, the main features of the tides using the theory of universal gravitation. These features included the following: spring-neap tides resulting from the gravity of both Moon and Sun with spring tides happening at syzygy; spring tides being larger still when lunar perigee coincides with syzygy; diurnal inequality occurring from the Moon and Sun being above or below the equator; solar tides being greatest in winter (perihelion); the anomalous diurnal tides in the South China Sea (see later); and at the end of Book 3, a calculation of the magnitude of tidal motions from first (mathematical)

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A journey through tides

principles. We have Edmond Halley to thank for paying for the publication of the Principia, the finances of the Royal Society having been strained by the publishing failure of the History of Fishes by Francis Willughby. We tend to think now that Newton’s explanations of gravity and tides were accepted rapidly after the publication of the Principia, but that was far from the case for at least the next half century (van Lunteren, 1993). In fact, The Gentlemen’s Journal in 1692 listed 10 competing explanations for the tides and made the reasonable statement that their proliferation was leading to confusion (Roos, 2001). Of course, Newton’s theory was promoted energetically by his supporters in the scientific world such as Edmond Halley, and it eventually became the “standard model,” but there was still to be more general reaction from other directions. Important in England was the Hutchinsonian movement named after John Hutchinson (1724–70) (Wilde, 1980; Aston, 2008). This was a loose collection of individuals in church and state who opposed the cultural dominance of Newtonian physics which, in their eyes, constituted the “Religion of Satan” (Fig. 1.6). Instead, they claimed that the truth lay in the original Hebrew text of the Old Testament. Hutchinson’s main personal objection to Newtonian philosophy was over the use of force as an explanatory concept without assigning a mechanical cause, an aspect of gravity which had concerned Newton himself (Aiton, 1969; Wilde, 1980). It would take a century after Newton’s Principia for the Hutchinsonian movement to die out.

5. Essays for the Acad emie Royale des Sciences In 1740, the Academie Royale des Sciences in France awarded four recipients with a prize for the best philosophical essay on the “flood and ebb of the sea” (Cartwright, 1999). One was Colin Maclaurin, Professor of Geometry at Edinburgh University, and another was Daniel Bernoulli, Professor of Anatomy and Botany at Basel. Maclaurin and Bernoulli can serve as examples of how well-connected scientists now were and of how their developments of Newton’s theory were to lead to practical improvements in the provision of tidal information. Maclaurin proved what Newton had assumed intuitively, that the shape of an otherwise spherical ocean in static equilibrium with the tidal force induced by a disturbing body (i.e., either the Moon or Sun) is a prolate spheroid (a shape like a rugby ball with one elongated axis of symmetry), the major axis of which points toward the body. Bernoulli’s Traite Sur le Flux et le Reflux de la Mer in effect extended Maclaurin’s essay, although at the

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Fig. 1.6 William Hogarth designed this frontispiece for a pamphlet against the Hutchinsonians in 1763. A witch sitting on top of a crescent moon is urinating a cascade onto the rocks below, on which there is a bound copy of “Hutchin” (i.e., the Moses’s Principia of 1724 and 1727 by John Hutchinson), and so drowning a group of black rats (i.e., the followers of Hutchinson). Some of the rats are vainly trying to gnaw at Newton’s philosophy, represented by a bound copy of “Newton” (i.e., his Principia) and a telescope. Pen and ink with gray wash. (© The Trustees of the British Museum.)

time he was unaware of Maclaurin’s contribution (Aiton, 1955b). His essay introduced the so-called Equilibrium Theory, which describes the temporal and spatial structure of the equilibrium tide due to the Moon and Sun in combination. In other words, Bernoulli combined the two individual prolate spheroids into one overall shape and introduced the lunar and solar orbits

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A journey through tides

Fig. 1.7 Bernoulli’s diagram for the parameters of the combined equilibrium tide due to the Sun (S) and Moon (M). The combined tide at point p is given by h (cos2θs 1/3) + H (cos2θm 1/3). For a practical tide table, one simply has to know the relative proportions of the solar and lunar semidiurnal tide at that location (h and H, respectively) with the angles θs and θm obtainable from the Nautical Almanac. (Adapted from a diagram in Cartwright, D.E. 1999. Tides: A Scientific History. Cambridge University Press: Cambridge. 292 pp., a similar diagram can be found in Aiton, E.J. 1955b. The contributions of Newton, Bernoulli and Euler to the theory of the tides. Ann. Sci., 11, 206–223, https:// doi.org/10.1080/00033795500200215.)

and Earth rotation into the discussion, so that the time dependence of the equilibrium tide at any point on the Earth’s surface could be parameterized (Fig. 1.7). Bernoulli had also found that Newton had overestimated the ratio between the lunar and solar tides; using French observations, he arrived at a value of 2.5, close to modern estimates (Ekman, 1993). As we know now, the spatial variation of the tide in the real ocean is much more complicated than that of the equilibrium tide because of the ocean dynamics, but Bernoulli found that its temporal variation at any location with predominantly semi-diurnal tides (which includes most of the European Atlantic coastline) can be parameterized in terms of the equilibrium tide to a good approximation with a small number of adjustments. As a result, he was able to compute a generic tide table for such locations (see Fig. 5.4 of Cartwright, 1999). An important factor with respect to the Bernoulli method was the publication of the Nautical Almanac under the direction of the fifth Astronomer Royal Nevil Maskelyne, who had his own interests in ocean tides. As is well known, the Nautical Almanac was published primarily for the purpose of navigation at sea using the method of “lunar distances.” However, the tables of lunar and solar parameters contained in the 1767 and subsequent editions were in an ideal form for application to Bernoulli’s method.

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Another requirement is the availability of some observations of heights and times of high waters at the location in question in order to make the necessary small adjustment to Bernoulli’s generic method. In practice, this involves a scaling factor for the average range of the tide at that location, an adjustment for its age of the tide (when one has maximum high water following a New or Full Moon), and an additional one for the relative importance of the Moon and Sun (which can be obtained from inspection of the real spring-neap variation in high waters). As a result, shortly after the first Nautical Almanac was published one finds the first reliable, publicly accessible tide table being produced for the port of Liverpool by Richard and George Holden, tuned up from Bernoulli’s generic method thanks to the availability of 4 years of observations by William Hutchinson, the Liverpool dockmaster. The Holden family tried to keep the details of their method secret for many years. However, it was finally shown by Woodworth (2002) to be simply a version of that specified by Bernoulli (Fig. 1.8). The important connections between the individual characters in this story can be demonstrated by reference to James Ferguson who was an astronomer, elected Fellow of the Royal Society in 1763. He wrote several astronomical treatises, of which one contains an “exercise” for the construction of a tidal clock (Ferguson, 1773). He made part of his living by travelling around the country and presenting lengthy series of lectures on scientific

Fig. 1.8 Heights of daytime high waters for 1795 from the Holden tables (dots) and as computed by the author using the Bernoulli method (line). (From Woodworth, P.L. 2002. Three Georges and one Richard Holden: the Liverpool tide table makers. Trans. Hist. Soc. Lancashire Cheshire, 151, 19–51.)

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subjects. He visited Liverpool on several occasions and usually stayed at Hutchinson’s house. Ferguson is believed to be the person who encouraged Hutchinson to begin his tidal measurements. He was a lifelong friend of Maclaurin whom he had first contacted on aspects of astronomy. Ferguson and Maclaurin were also linked through Murdoch Mackenzie, a native of the Orkneys and a “much travelled marine surveyor.” Mackenzie had started his career in hydrography on Maclaurin’s advice, and Maclaurin had recommended him for the task of surveying the Orkneys, where he made his own observations on the tides (Mackenzie, 1749). Ferguson and Mackenzie met in Edinburgh and were very close, Ferguson naming his third son after Mackenzie, and both being called proteges of Maclaurin (Millburn, 1988). Mackenzie is thought to have been the person who introduced Ferguson to Hutchinson. In turn, Hutchinson was either a friend or close colleague of Richard Holden, as demonstrated by their common interests in astronomy and the invention of lighthouse reflectors (Woodworth, 2002). While the essays of Maclaurin and Bernoulli can be seen to have led to practical benefits, those of Leonhard Euler, Professor of Mathematics at St. Petersburg, and Antoine Cavalleri, Professor of Mathematics at Cahors, were in retrospect less useful. However, all four essays could at least be said to have the merit of having learned from what had come before. Euler showed that it was the horizontal, and not vertical, component of the force field which leads to tidal motion (Aiton, 1955b). Cavalleri’s essay built on the work of Descartes, although he disagreed with the earlier theory because of its lack of a major contribution from solar tides. He also disagreed with Newton’s theory of gravitation and instead persevered on a fruitless development of the Cartesian theory of vortices. Aiton provides a detailed discussion of the vortex theory of planetary motions (Aiton, 1957, 1958a, 1958b). He remarks that Cavalleri had nothing really new to add to this subject (Aiton, 1958b). One might note that the essays on the tides were not the only ones at this time. For example, both Bernoulli and Euler had essays on magnetism in 1746. Bernoulli was awarded the grand prize of the Paris Academy 10 times in all, and Euler 12 times, for essays on various topics (Fara, 1996). Maclaurin was arguably also the first to identify what is now called the Coriolis Effect (Harris, 1898).

6. Before and after Newton In this chapter, I have tried to present the contrasting approaches to ideas on the tides in the periods before and after Newton. Theories in the earlier

Tidal science before and after Newton

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period tended to be little more than hand-waving speculations with an absence of mathematical or any other rigor. Just half a century later, as we have demonstrated by the Paris essayists (especially Bernoulli and Maclaurin but all four to some extent), there was a greater willingness to base theories on observations and to learn from the earlier work of others. And in Bernoulli’s case, a major benefit of the research was the generic tide table capable of application to anywhere in the world with a semidiurnal tidal regime. Of course, such a comparison could be said to be a false one given what the second set would inevitably have learned from Newton. Nevertheless, the contrast is quite apparent. However, we have shown that acceptance of Newton’s theories was not universal and immediate. In addition, it is disappointing that after the achievements of Newton and Halley, investigations of tides became largely a continental European and not English pursuit, culminating at the end of the century in the dynamical tidal theory of Laplace with tides considered as fluid in motion on a rotating Earth. Laplace’s Traite de Mecanique Celeste, written in five parts between 1798 and 1825, can be regarded as almost as important to the study of tides as Newton’s Principia. In fact, it has pointed out that the Laplace Tidal Equations (Laplace, 1775, 1776) can be regarded as the first formulation of an ocean model, in this case a tide model (Arbic, 2022). Meanwhile, in England, there was a “doldrums of UK tidal science” until the work during the 19th century by the UK scientists mentioned in the Conclusions later (Rossiter, 1971). Initiatives in tidal measurement, as well as in tidal theory, passed to continental Europe (especially France) after Newton. Notably, the times and heights of high and low waters were recorded at Brest between 1711 and 1716 which were sent for analysis by Jacques Cassini at the Academie Royale des Sciences. Later measurements were also made at Brest and neighboring ports. Cassini interpreted these data as support for the Descartes theory of the tides. Cartwright (1972) and W€ oppelmann et al. (2006) discuss their use in modern analyses. Extended measurements of high waters in England had to await those of William Hutchinson at Liverpool in 1764–93. The practice of forgetting findings of earlier investigators was repeated through the years. For example, we have shown that some areas of the ocean were known by the investigators in the ancient world to have diurnal, rather than semidiurnal tides. It was remarked on subsequently regarding locations far from Europe. One case concerns the communication by Francis Davenport of the East India Company to the Royal Society in 1678, referring to the anomalous diurnal tides of the Gulf of Tonkin (South China Sea).

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The matter was taken into consideration by Halley in 1684, after a delay of 6 years, inconveniently to some extent just before the publication of the Principia (Cartwright, 2003; Hughes and Wall, 2006). However, tides continued to be understood to mariners of powerful north-western European countries in the 18th century in terms of only the two parameters suitable for describing the predominantly semidiurnal tide: rise and fall, which was essentially twice the tidal amplitude, and high water full and change, which would later be known as establishment. The latter parameter is the lag between the Moon’s transit of the meridian at the location in question and the next occurrence of high tide, at times when the Moon, Earth, and Sun are aligned (syzygy). Such information was to be collected and made available in the tables of Lalande (1781). Therefore, the mariners carried this picture of the tide with them when they embarked on their wider voyages of exploration. For example, Woodworth and Rowe (2018) discuss how puzzled James Cook was by diurnal inequality in the tide along the Queensland coast, a factor which led to the near sinking of the Endeavour in June 1770. One might have thought that the possibility of diurnal tides at distant locations would have been known to most captains by the time of Cook’s voyage a century later than Davenport’s report. The Holden tide table makers at around this time certainly knew of the night-time tides at Liverpool being lower than day-time ones for November-April (and vice versa), primarily due to the local phase lag of the K1 diurnal constituent with an amplitude of 11 cm, and made an appropriate adjustment for their “Bernoulli predictions” (Woodworth, 2002). Diurnal inequality was later to be an important aspect of tidal research by Whewell and others in the 19th century.

7. Conclusions Newton has been represented in this chapter as a separation between one era in the history of tidal science and the start of another (with admittedly a number of omissions such as the work of Halley). However, with the benefit of hindsight, one can identify other eras during the following centuries. These include the dynamical work of Laplace in the late 18th and early 19th centuries; the development of the harmonic method by Kelvin and Darwin, the drawing of global cotidal charts by Whewell, Airy, Bache, Harris, and others, and the technological developments of tide gauge networks (the data from which are now important for long-term climate studies) and

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tide prediction machines (Woodworth, 2020) during the 19th century; and the theoretical work of Proudman and developments in tidal prediction by Doodson at the Liverpool Tidal Institute (LTI) in the mid-20th century (Woodworth et al., 2021). It is obvious that there are many omissions and simplifications in such a list. Nevertheless, it seems that when Cartwright produced the manuscript for his book in 1996, he thought that he was marking the end of another era (Chapter 1, page 4). The postwar decades of the 20th century had seen the deployment of bottom pressure recorders for tidal measurements in many parts of the ocean (in which Cartwright himself had been a leading participant), and the publication of comprehensive global tidal models involving many constituents, notably the model of Schwiderski (1980). However, Cartwright (1999) also noted that research on tides was hardly at an end, especially given the potential provided by satellite altimetry and other technologies. This was already made clear at a meeting at the Royal Society in 1996 to celebrate his 70th birthday, with many papers subsequently published in a special issue of Progress in Oceanography (Ray and Woodworth, 1997). The TOPEX/POSEIDON and JASON series of satellite altimetry missions has since revolutionized the development of regional and global tide models (Stammer et al., 2014), and for some tidal researchers, this is very much the “age of altimetry.” However, there is still a need to understand the tides better in coastal waters and at high latitudes, and the altimeters of recently launched and upcoming missions such as CryoSat-2, Sentinel-3, Sentinel-6, and SWOT should meet these challenges to a great extent. Tides continue to be important factors in a wide range of research that is of both scientific and practical importance, as noted by the various papers in a special issue of Ocean Science to mark the centenary of the founding of the LTI in 2019 (Woodworth et al., 2021). Cartwright himself had been Assistant Director at the Institute of Oceanographic Sciences (Bidston), as the LTI was known at the time. It seems that the ocean tides will never cease to fascinate. Ideas for alternative theories of the tides continue to appear on a regular basis; Doodson and Warburg (Admiralty Manual of Tides, 1941) remarked that “There are few subjects which have been more associated with fantastic theories and speculations.” But that is fine, each theory and speculation presents an intellectual challenge, and it is always possible that new perspectives may be obtained by discussing them. Of course, the essential aspect of any new theory should be that it explains all available data as well as an existing theory and in addition comes up with predictions that differ from the earlier theory that can be tested by measurement (the Baconian Method).

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In summary, I hope this article has interested the reader enough to purchase David Cartwright’s book for an excellent treatment of the history of research into the tides through the years. To avoid major overlap with the chapters in his book, I have tried to include as many relevant references as possible since that book was published. For example, there is now an extensive amount of information in Wikipedia and elsewhere on the internet (which Cartwright would probably not have regarded as respectable). Otherwise, I would recommend the reader to consult Harris (1898), the “concise history” of Ekman (1993) and the many papers of Aiton in Annals of Science.

Acknowledgments I thank Mattias Green and Joa˜o Duarte for the invitation to write this chapter and David Pugh and Chris Hughes for providing useful comments on it. I am grateful to John Glassie for pointers on Athanasius Kircher. Some information on the Holden tide tables in this chapter was adapted from Woodworth (2002).

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Wikipedia, 2022a. https://en.wikipedia.org/wiki/Theory_of_tides. Wikipedia, 2022b. https://en.wikipedia.org/wiki/John_of_Wallingford_(d._1258). Wikipedia, 2022c. https://en.wikipedia.org/wiki/Johannes_Kepler. Wikipedia, 2022d. https://en.wikipedia.org/wiki/Galileo_Galilei. Wikipedia, 2022e. https://en.wikipedia.org/wiki/Discourse_on_the_Tides. Wikipedia, 2022f. https://en.wikipedia.org/wiki/Francis_Bacon. Wikipedia, 2022g. https://en.wikipedia.org/wiki/William_Gilbert_(physician). Wikipedia, 2022h. https://en.wikipedia.org/wiki/Athanasius_Kircher. Wikipedia, 2022i. https://en.wikipedia.org/wiki/Republic_of_Letters. Wikipedia, 2022j. https://en.wikipedia.org/wiki/Mundus_Subterraneus_(book). Wikipedia, 2022k. https://en.wikipedia.org/wiki/Rene_Descartes. Wilde, C.B., 1980. Hutchinsonianism, natural philosophy and religious controversy in eighteenth century Britain. Hist. Sci. 18, 1–24. https://doi.org/10.1177/007327538001800101. Woodworth, P.L., 2002. Three Georges and one Richard Holden: the Liverpool tide table makers. Trans. Hist. Soc. Lancashire Cheshire 151, 19–51. Woodworth, P.L., 2020. Tide prediction machines at the Liverpool tidal institute. Hist. Geo Space Sci. 11, 15–29. https://doi.org/10.5194/hgss-11-15-2020. Woodworth, P.L., Rowe, G.H., 2018. The tidal measurements of James cook during the voyage of the Endeavour. Hist. Geo Space Sci. 9, 85–103. https://doi.org/10.5194/ hgss-9-85-2018. Woodworth, P.L., 2022. Advances in the observation and understanding of changes in sea level and tides. Ann. N. Y. Acad. Sci. https://doi.org/10.1111/nyas.14851. Woodworth, P.L., Green, J.A.M., Ray, R.D., Huthnance, J.M., 2021. Preface: developments in the science and history of tides. Ocean Sci. 17, 809–818. https://doi.org/ 10.5194/os-17-809-2021. W€ oppelmann, G., Pouvreau, N., Simon, B., 2006. Brest Sea level record: a time series construction back to the early eighteenth century. Ocean Dyn. 56, 487–497. https://doi. org/10.1007/s10236-005-0044-z. Wright, J., 1923. The science of Herodotus. Sci. Mon. 16, 638–648. https://www.jstor.org/ stable/6327.

CHAPTER 2

Introducing the oceans Yueng-Djern Lenna, Fialho Nehamab, and Alberto Mavumeb a

School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Escola Superior de Ci^encias Marinhas e Costeiras, Universidade Eduardo Mondlane, Quelimane, Mozambique b

1. Our blue planet The dawn of the space age not only heralded humanity’s exploration beyond our planet but also provided the unprecedented opportunity to look back from afar and consider our planet and how it sustains our existence. The view of Planet Earth from space (Fig. 2.1), as immortalized by the Apollo 17 crew in 1972, brings our home into sharp focus as a beautiful blue planet against the infinite blackness of space. Thus far, our exploration of space by manned and unmanned spacecraft, as well as with our powerful radio telescopes, has yet to locate and identify another planet so abundant in the surface waters that make our planet blue. Any oceanographer can tell you that our oceans cover over 70% of the Earth’s surface and are indescribably fascinating and beautiful, but perhaps more importantly, are absolutely vital to life on Earth and the functioning of our climate system. Many different types of water bodies exist on the surface of the Earth, from lakes fed by melting mountain snowpacks to rivers and ponds. To follow a river onwards from its source will almost inevitably lead you out to sea, to the big blue ocean that begins where the land ends. Our oceans reach from pole to pole, covering vast areas with seawater that moves in myriad ways. Not all seawater resides in liquid form; some of it is frozen into sea ice in the far north and the far south, floating on the polar seas as drifting ice packs. The oceans reach far deeper from the sea surface (down to 11 km in Challenger Deep of the Marianas Trench) than the tallest mountains rise above the sea level (Fig. 2.2). Yet paradoxically, typical depths of ocean basins are shallow (under 4 km on average, reaching 5 km at the abyssal plains) relative to their horizontal spread (spanning tens of thousands of kilometers between continents) such that the oceans are essentially a thin blue layer lying atop the Earth’s crust with an aspect ratio akin to a sheet of paper.

A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00018-2

Copyright © 2023 Elsevier Inc. All rights reserved.

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Fig. 2.1 Blue marble. View of Planet Earth from the window of Apollo 17 as photographed by the crew members in December 1972. (Photograph from NASA.)

Within this thin blue layer lives the oceanic ecosystem, where the seaweed and phytoplankton that perform photosynthesis in the sunlit upper ocean provide 50% of the oxygen on Earth while fixing dissolved atmospheric carbon dioxide into sugars (Friedlingstein et al., 2020). Predators that consume phytoplankton and descend to respire at depth, or death and other biological processes that result in sinking organic carbon detritus known as marine snow, in addition to physical processes like downwelling currents conspire to store, within our oceans, 50 times the carbon found on land (Friedlingstein et al., 2020). Consequently, research into ocean sequestration of atmospheric carbon dioxide and other ways in which the oceans are currently holding back the worst impacts of climate change continues unabated, and with increasing intensity in the early 21st century. Our oceans’ contribution to global carbon cycles aside, their impact on the day-to-day lives of people is equally profound. Approximately, 40% of humanity lives within 100 km of the coast (Small and Nicholls, 2003), with many depending directly on it for food, livelihoods, and recreation, and being subject to marine-mediated impacts on local weather. These coastal populations are among the fastest growing on the planet and many such

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Fig. 2.2 Bathymetry of the world’s ocean mapped on the Spilhaus projection. Darker blues indicate deeper waters.

populations, particularly across south and southeast Asia, face increasing vulnerability to extreme weather events such as coastal flooding exacerbated by sea level rise (Neumann et al., 2015). In addition, global commerce and our economies rely on shipping to transport goods and products across the oceans. For instance, at least 70% of all trade between Europe and other nations takes place by sea, utilizing the global shipping network that connects all the coastal nations of our globe (Hoffmann et al., 2017). The ocean’s relevance to how all of us live cannot be overstated. Because of its relative remoteness and unsuitability as an easily-settled human habitat, the sea remains a place of mystery and romance, both scientifically and literally. We have yet to truly plumb the depths to discover all there is to know about the ocean and the inhabitants of this watery realm. As Helen Scales so evocatively explains in her popular science book The Brilliant

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Abyss (Scales, 2021), never-before-seen deep sea species continue to be discovered on every new deep ocean expedition that is mounted. Consequently, marine biologists consider our understanding to be barely scratching the surface of all there is to know about our fellow earthlings. From tales of the lost Atlantis to Jules Verne’s 20,000 Leagues under the Sea (Verne, 1869) and so on, the sea remains an infinite source of inspiration for storytellers, explorers, and scientists. Nonetheless, there is a lot we can say about how our oceans work. We know, for instance, that our sun is the primary source of heat for our planet’s surface and our oceans. We know that our atmosphere, land, and oceans absorb this solar radiation at variable rates and that the solar radiation is unequally distributed from the equator to the poles. This spatial variability in solar heating and the intrinsic rotation of the Earth together drive atmospheric circulation, which in turn forces ocean circulation. Ocean currents themselves are subject to angular momentum conservation as they flow around our rotating Earth, and from the subtropics to the poles, we experience seasonal variability that results from Earth’s axis of rotation being nonnormal to our orbital plane around the sun. Speaking of which, both the Sun and Moon exert a gravitational pull on our oceans that drive the tides, which are the focus of this book. This chapter will introduce fundamental oceanography concepts that will help you understand tidal dynamics and the impact the tides have on our oceans and planet that is discussed in the other chapters. Here, we explain how seawater is characterized by its state variables like temperature and salinity and how these seawater properties tell us about ocean mixing and dynamics. We will explain how our globally interconnected ocean circulates within gyres residing within the major ocean basins, which in turn are surrounded by continental shelf seas that creep up estuaries to mix with freshwater from rivers, and how it freezes in the polar regions to float about the sea. We will consider how seawater properties act as tracers that tell us where and when any specific water parcel was last at the surface of the ocean and how these water masses spread throughout the global ocean. Finally, we will explain the ocean’s role in Earth’s climate system and how it mediates our weather and enables life on Earth.

2. Physical properties of seawater Energy input to the ocean via winds or tides drives ocean currents, but ocean currents can also be driven by differences in density. In the classic lockgate

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Fig. 2.3 Lockgate schematic. Left: Lockgate closed configuration with a partition separates fluids of different density. Right: partition is removed as the lockgate is “opened,” allowing the dense water to flow under the light water as the water column adjusts.

experiment (Fig. 2.3), dense water is separated from lighter water in a rectangular tank by a plane barrier, representing a situation that is typical for regions of freshwater influence, but is also known to occur in open ocean when there are density gradients. When the lockgate is opened by removing the barrier, the dense water slides under the lighter water as a density current, eventually filling the bottom of the tank beneath a lighter layer. This adjustment from the initial high potential energy state to a stable two-layer minimum potential energy state is driven by the density differences between the two layers. Seawater density (usually denoted by ρ) is not only a driving force in density currents but is also a key property to the understanding of how seawater is layered, or stratified, within the global ocean. Ocean stratification in turn sets the kinetic energy input required to overcome the potential energy barrier to mix different layers and their respective properties such as heat, salt, and nutrients together. The physical properties that determine seawater density are known as state variables, and these are the quantities measured by oceanographers seeking to understand ocean physics. The state variables of seawater that set its density are temperature, T, and salinity, S. Temperature is a measure of the heat stored within the ocean and increases in heat content result in the thermal expansion of seawater, reducing its density. Salinity is a measure of the salt content of the seawater and is the counterpoint to its freshwater content. Salinity is defined as the concentration of salt (typically dissolved sodium chloride) in seawater and is given in parts per thousand by weight or grams per kilogram. Increases in salinity imply higher concentrations of dissolved salt and thus increase the density of seawater. The relative impact of changes on seawater density are given as Δρ ¼ αΔT + βΔS, where α is the coefficient of thermal expansion and β is the haline contraction coefficient, approximately equal to 0.0008. The thermal expansion coefficient is dependent on the temperature, and we do not have a simple expression for it.

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The heat content that sets seawater temperature mostly comes from the Sun. Solar radiation across the visible spectrum, from the short ultraviolet/ blue wavelengths to the longer red wavelengths, easily passes through the Earth’s atmosphere without much absorption. When it reaches the ocean, the Sun’s rays penetrate the surface ocean powering photosynthesis in marine ecosystems and illuminating the upper ocean. This radiation does not reach the seafloor of ocean basins. Indeed, the red wavelengths are quickly absorbed by, and warm, water molecules such that by about 30 m below the sea surface, blue light dominates the view. Descend much further, and soon even these short wavelengths are absorbed by the water as the ocean grows as dark as midnight. All this absorption of solar radiation warms our oceans from the top, resulting in sea surface temperatures ranging from approximately 2 °C in polar oceans to 30 °C at the equator (Fig. 2.4, lower panel). Consequently, any profiling temperature-measuring instrument dropped into the subtropical or mid-latitude ocean will register everdecreasing temperatures as it descends (Fig. 2.5). In addition to solar radiative fluxes, the ocean can also gain heat through conductive fluxes (e.g., hot atmosphere over cool ocean), warm river discharge into the polar oceans (Park et al., 2020), and be heated by hydrothermal vents from below (Adcroft et al., 2001). Conversely, the ocean also loses heat at the surface through conduction (sensible heat fluxes), emission of longwave infrared radiation, and latent heat fluxes associated with phase changes arising from evaporation and ice formation/melting. High winds and the breaking of wind-driven waves will tend to enhance evaporation and ocean heat loss to the atmosphere. The combined effect of these processes effectively sets the ocean temperature, which is the main determinant of ocean density changes outside of the polar oceans. Salinity of course arises from dissolved salt put into the oceans. The total salt content of the oceans is thought to have accumulated over millennia from mineral rock salts dissolved and washed into the sea by the rain. Rainwater-dissolved mineral rock salt does continue to enter the ocean from rivers; however, these modest increases in salt content are more than offset and diluted by the freshwater input of the river water itself. As a result, the salt content of the ocean is considered to be relatively stable at ocean salinities of about 35 parts per thousand, which is found almost everywhere in the global ocean. Nonetheless variations in ocean salinity do exist, and these variations are largely driven by the addition or removal of freshwater from the ocean. Along coastlines, freshwater input from rivers results in regions of freshwater

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Fig. 2.4 Average sea surface salinity (upper) and sea surface temperature (lower) for the period 1930–2020, from the hydrographic ARMOR3D dataset (Guinehut et al., 2012).

influence where mainly fresh regions are separated from seawater by tidal mixing fronts (Sharples and Simpson, 1993). Farther out to sea, it was Robert Boyle who first noted that the ocean was salty from the sea surface to the seabed, and that it is the competing effects of evaporation and

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Fig. 2.5 Temperature profiles at (A) high, (B) mid, and (C) low latitudes.

precipitation that determined sea surface salinities (Boyle, 1685). Thus ocean salinity is generally higher at the surface than at depth because only the surface is subject to evaporation (Fig. 2.6). Across different latitudes, precipitation under the intertropical convergence zone and mid-latitude storm tracks, evaporation in the sunny subtropical latitudes, and sea ice and glacial

Fig. 2.6 Salinity profiles at (A) high, (B) mid, and low latitudes for the Atlantic and Pacific oceans.

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melt in the polar oceans result in alternating patterns of sea surface salinity highs and lows (Fig. 2.4, upper panel). Partially because of the very small range of ocean salinities (i.e., 34–37), but mainly because of the strong dependence of density on temperature, the ocean salinity impact on density is generally minor in the global ocean. This impact is larger in semienclosed basins and estuaries and may even surpass that of the temperature. At most latitudes, the twin effects of temperature and salinity result in a warm and often salty surface mixed layer (SML, see T-S profiles). Within the surface mixed layer, generally 50–100-m deep except in winter when the SML at high latitudes reaches the 400 m depth, wind-driven mixing and convection arising from surface buoyancy fluxes combine to thoroughly mixed heat and salt properties. Below the largely homogenous mixed layer lies, in general, the ocean thermocline. This thermocline can be several hundred meters to a km thick and is characterized by a steep vertical gradient in temperature. Where temperature effects on density dominate (i.e., most of the global ocean), the ocean thermocline functions also as its pycnocline in which steep gradients in density act as energy barriers to mixing of water properties between the layers above and below the pycnocline. Below the SML, both ocean temperatures and salinities generally decrease with depth, with the density reduction from lower salinities being greater than the offset by the contraction due to cooler temperatures. Thus the ocean is usually stably stratified with lighter waters overriding dense waters. At very low temperatures, the salinity influence on density takes precedence over temperature. As seawater cools toward the freezing point, it contracts and its density increases until it reaches the temperature of maximum density (Fig. 2.7). For pure water, this is 3.98 °C, and for typical seawater salinities, it is approximately 0 °C. As water cools below the temperature of maximum density, liquid water expands again until it reaches the freezing point and changes phase into ice which is even less dense and floats atop the liquid water. Thus in the polar oceans, at near-zero temperatures and below, the density and the resulting stratification is set by salinity, rather than temperature. This is why, for instance, the Arctic is considered an “upsidedown” ocean, where a very fresh cold surface mixed layer floats above a halocline layer, characterized by a marked vertical gradient in salinity, which in turn overrides much warmer saltier water of Atlantic origin below, a reversal of the water properties’ depth profile seen at lower latitudes. The last important state variable of seawater is its in situ pressure, P. Seawater contracts, slightly, under pressure, and within the ocean, this hydrostatic pressure is given by P ¼Hρg, where H is the water depth and

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Fig. 2.7 Temperature of maximum density of seawater at P ¼ 1.023 105 Pa (left) and water masses displayed on a T-S diagram (right).

g is the acceleration due to gravity. Hydrostatic pressure is isotropic causing the seawater to contract equally in all directions, such that the increased proximity of water molecules effectively increases the internal energy or heat of the water parcel under pressure. Hence, the pressure effect on temperature must be accounted for in order to be able to directly compare the temperature of water parcels at different depths. Therefore pressure, or depth by inference, is always measured alongside temperature and salinity. Historically, once the pressure effect has been accounted for, the corrected in situ temperature T is known as potential temperature, θ, which is defined as the temperature of the water parcel if it had been adiabatically moved to a reference depth, usually the surface. Using the modern standard TEOS10 (Thermodynamic Equation of State 2010, http://www.teos-10.org) Gibbs seawater routines based on the thermodynamic principles, oceanographers have adopted conservative temperature, Θ, and absolute salinity, S, as the new standard pressure-corrected hydrographic state variables.

3. Geography and ocean circulation On World Oceans Day, June 8, 2021, the National Geographic Society acknowledged the Southern Ocean as the fifth ocean basin on our planet. The Southern Ocean joined the Atlantic, Pacific, Indian, and Arctic Oceans in recognition of its unique circulation and water properties that distinguished it from the other basins. This of course was not remotely news to oceanographers who have long considered the Southern Ocean as a major ocean basin, that yes, is unique in many respects. The Southern Ocean hosts

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the mighty Antarctic Circumpolar Current (ACC) which rushes in a continuous braided eddying stream around Antarctica, connecting the Atlantic, Pacific, and Indian Oceans at high southern latitudes. The ACC is forced directly by westerly winds, which are unimpeded by land masses as they encircle the globe, and effectively separates warmer subtropical waters to the north from colder polar waters to the south (Speer et al., 2000). These winds also drive a northward surface Ekman transport that draws isopycnals (i.e., surfaces of equal density) toward the surface, making the Southern Ocean the largest upwelling region on the globe (Fig. 2.8). This upwelling brings nutrients (i.e., nitrates, phosphates, silicates) to the upper layers of the ocean and has profound consequences for water mass formation. Along with the nutrients, CO2-enriched deep waters are vertically imported to the upper ocean, allowing them to interact with the atmosphere and impact the climate (a topic that will be discussed in the next two sections).

Fig. 2.8 Southern Ocean upwelling and downwelling indicated by green arrows against a background of potential temperature. Upper and lower circumpolar deep water (UCDW, CDW, LCDW) upwell along isopycnals, while Antarctic Bottom Water (AABW) cascades down the continental slope to the abyss. Subantarctic mode water (SAMW) and Antarctic Intermediate Water (AAIW) are carried north by the wind-driven Ekman transport and subduct below the subtropical mode waters to the north. (Modified with permission from Silvester, M.J., Lenn, Y.D., Polton, J., Rippeth, T. P., Morales Maqueda, M. 2014. Observations of a diapycnal shortcut to adiabatic upwelling of Antarctic Circumpolar Deep Water. Geophys. Res. Lett. 41, 7950–7956.)

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Winds are also responsible for driving the circulation of the five major subtropical ocean gyres. These gyres occupy the 4–5-km-deep basins of the North and South Atlantic, the North and South Pacific, and the Indian Oceans. Moving either north or southward away from the equator, the prevailing wind direction switches from the easterly trade winds blowing along the equator to westerly winds as latitude increases. This divergence in the zonal wind stress drives a surface Ekman convergence, piling water into the center of each subtropical gyre, and depressing the thermocline (Fig. 2.9). Ocean dynamics within subtropical gyres are dominated by the Sverdrup balance between the Earth’s rotation and wind stress, such that the depth-integrated meridional basin transport V is given by βV ¼ k  r  τ

(1)

where β is the meridional gradient of the Coriolis parameter f, k is the unit vector in the z (vertical) direction, and τ is the surface wind stress

Fig. 2.9 Sverdrup gyre cross-section, reproduced from Talley et al. (2011). Wind stress (red arrows) drive a basin-scale horizontal circulation (blue arrows) and Ekman transports (black arrows) that result in a depressed thermocline in the middle of subtropical gyres.

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(Sverdrup, 1947). This broad equatorward flow across most of the subtropical gyres effectively loses planetary vorticity as it nears the equator. To balance the mass redistribution of this southward flow, seawater must be returned polewards in narrow, intensified western boundary currents where vorticity is gained as a result of the coastal friction onshore imposing a zonal velocity gradient on the currents (Stommel, 1948). Within these western boundary currents, current speeds can reach up to 1 m s1. The key western boundary currents are the Gulf Stream and Brazil Current in the Atlantic, the Agulhas Current of the Indian Ocean, and the Kurushio and East Australian Currents of the Pacific. Poleward transport of warm salty waters in these western boundary currents vary from the 10 Sv of the Brazil Current (Stramma et al., 1990) to the 75 Sv of the mighty Agulhas Current (Beal and Bryden, 1999). Away from western boundaries, ocean currents within the central gyres tend to be much smaller, typically a few cm s1. These recirculating subtropical gyres are key components of global ocean circulation in which the western boundary currents, in particular the Gulf Stream of the North Atlantic, play an outsized role in the transport of ocean heat and salt away from the equator. When tides traverse the deep ocean basins, their propagation speeds add to the ambient gyre circulation. However, over the deep basins, the tidal gravitational pull of the barotropic tide is spread out over the full water column resulting in very modest tidal currents of a few cm s1. The same gravitational pull exerted over the much shallower continental shelf seas will result in much bigger tidal currents, sometimes exceeding 50 cm s1 in places such as the Patagonian shelf (Glorioso and Simpson, 1994) or 100 cm s1 on the European shelf. Continental shelf seas mark the transition zone from the deep ocean basins to land and may be up to 4–500 m deep, although most shelf seas are closer to 100-m deep. Continental shelf seas are themselves separated from the deep ocean basins by continental shelf breaks, where the seabed typically slopes very steeply away from shore. Continental shelf breaks are energetic regions in which strong, generally narrow along-slope currents (Huthnance, 1984) act as barriers dividing the basin waters from the shelf waters. Potential vorticity constraints associated with changes in the water column thickness also conspire to limit shelf-basin exchange. Yet, shelf breaks are known to be active mixing regions where internal tides break and dissipate their energy and bed friction increases rapidly onshore (Chapter 5). Up on the shallow continental shelf seas themselves, kinetic energy input by winds dissipated at the base of a mixed layer effectively mixes much of the

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water column than in the deep ocean. Both wind and tidal energy input can be dissipated either by shear-driven mixing at the base of seasonal summer mixed layers (Rippeth, 2005) or bottom friction (Simpson et al., 1996). Winter cooling of surface waters increases their density, destabilizing the water column, driving convection that contributes to the mixing from winds and tides. Consequently, in the winter, continental shelf seas are typically characterized by homogeneous water columns, whereas warming in spring and summer leads to the development of a warm surface mixed layer that persists into early fall. The development of seasonal stratification and the wind and tide-driven mixing across the continental shelf seas have two distinct and noteworthy impacts. On one hand, mixing allows the productive surface mixed layer to be frequently supplied with deep nutrients required to sustain primary productivity. And, on the other hand, the summer stratification assures that carbon exported below the seasonal thermocline in the forms of organic detritus or respiration is effectively separated from the surface and can be deposited on the seabed or exported off the shelf into the deep ocean basins. As a result, continental shelf seas which cover only about 7% of the globe, account for 20%–50% of the organic carbon stored by the global ocean (Thomas et al., 2004; Tsunogai et al., 1999). The productivity of continental shelf seas also supports rich ecosystems that provide vital commercial and nutrition resources for coastal communities. Tides propagating across continental shelf seas certainly do more than drive vertical mixing, and Chapter 11 will explore their impact on coasts in greater detail. For now, it is useful to know that asymmetries in tidal currents or interactions with topography can drive residual currents that contribute to the background circulation in a shelf sea (Polton, 2015). Tides are also critical in dispersing freshwater from rivers across the continental shelf beyond the estuary from which they emerge (Simpson et al., 1990). Estuaries are the regions in which the riverine freshwater first encounters seawater. In most estuaries, salty seawater sneaks landwards below the fresh and much lighter layer of river water heading out to sea (Fig. 2.10), creating a two-layer circulation that ebbs and flows with the tide (Knudsen, 1900). The dynamic balance of the two-layer estuarine circulation depends on physical factors such as river discharge, the salinity difference between the river and seawater, lateral salinity gradients, and geographical factors such as channel breadth and depth (Pritchard, 1956). In some instances, instead of the continental shelf sea exchanging saltwater for fresh in an estuary, we have the reverse exchange. This usually occurs where the river input is negligible relative to the high evaporation taking place within these

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Fig. 2.10 In an estuarine salt wedge, freshwater runs atop saline water.

“reverse estuaries.” Seawater entering reverse estuaries such as Spencer Gulf, South Australia, and the Mediterranean Sea, are subject to high evaporation on their circuit through the reverse estuary and lose buoyancy before exiting as an even saltier dense overflow below the incoming, comparatively fresher seawater (Fig. 2.11). Freshwater also plays an important role in the dynamics of the polar seas. Antarctic sources of freshwater stem from the glaciers and ice sheets extending out into the ocean. Large icebergs that break off from the glaciers or ice sheets can be advected far out to sea where they melt slowly releasing not just the freshwater that can exceed local precipitation-minus-evaporation fluxes (Silva et al., 2006) but also all the minerals and rocks entrained into the ice as the glacier scoured its bed on route to sea (Raiswell et al., 2008). In the far North, Arctic rivers drain vast areas that are home to 40 million people and input 11% of total global riverine freshwater input into approximately 4% of the area covered by oceans (Lammers et al., 2001), making this region the freshest ocean on the planet (Fig. 2.11). Small Arctic glaciers also contribute to the Arctic freshwater budget. All this freshwater insulates the warmer saltier waters below from the atmosphere, limiting the air-sea fluxes between the warm waters of Atlantic and Pacific origin and the atmosphere. In both polar regions, the seasonal cycle in sea ice formation and melting also exerts a big influence on the ocean dynamics. In winter, as seawater begins to freeze, it expels approximately 60% of its salt (Lake and Lewis, 1970). This process is known as brine rejection, in which near-freezing point high salinity water is fluxed into the water column below as freeze onset sets in. Brines continue to drain out of the predominantly fresh sea ice matrix throughout its lifespan (Niedrauer and Martin, 1979). Sea ice itself, particularly when consolidated into a drifting ice pack at 100%

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Fig. 2.11 Zonal section along 36 N with the Strait of Gibraltar to the right of the figure. Note that the high-salinity plume extends westward at 1000-m depth, indicating the presence of the Mediterranean Overflow Water (MOW).

concentration, acts as a barrier to fluxes of heat, freshwater, and momentum between the atmosphere and ocean (Green and Schmittner, 2015). In summer, when the sea ice melts after drifting around the ocean to places usually far removed from where it froze in the first place, the meltwater forms a freshwater cap over the ocean below. Spatial variability in the ocean stratification will have consequences on how the oceans respond to the surface wind stresses. Even though convection and two-layer exchange flows are mediated by density differences, it is still the wind that largely sets the basin-scale circulation patterns of the oceans. Beyond the five subtropical gyres (Fig. 2.12), smaller basins such as the subpolar Labrador Sea, Weddell Sea, and the Arctic Beaufort Gyre all feature their own wind-driven gyre circulations, while river plumes on

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Fig. 2.12 Schematic of the five gyres of the North and South Pacific, North and South Atlantic, and Indian Ocean are shown in red. The subpolar Labrador Sea circulation and polar Arctic Beaufort gyre are also shown in dark blue.

continental shelves are subject to the competing effects of the winds, conservation of vorticity, and tidal straining (Horner-Devine et al., 2015; Nehama and Reason, 2021).

4. Key water masses and global distributions As we have already seen, denser heavier water will slide beneath water that is lighter, more buoyant. In the ocean, this results in a layering of water of different densities that we refer to as stratification. It turns out that each layer, or water mass, has a distinct combination of temperatures and salinities that were set when it was last in contact with the surface. The process of surface properties reaching the deep ocean is known as ocean ventilation. Generally

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Fig. 2.13 Monthly mean meridional section along 25 W in the Atlantic Ocean for August 2020, plotted from the EN4 hydrographic dataset (Good et al., 2013). Salinity (top) and temperature (bottom) are overlaid with salinity contours in white. AABW, Antarctic Intermediate Water (AAIW), North Atlantic Deep Water (NADW), and MOW are indicated.

speaking, temperature and salinity can be used to track water masses as they are dispersed across the globe from their formation regions (Fig. 2.13). This section describes the major water masses involved in global circulation of our oceans. Clearly, the temperature, salinity, and density of a water mass will depend on where on the planet it was formed. In the middle of subtropical gyres, solar heating and evaporation result in warm salty subtropical surface mixed layers. Solar heating peaks in the summer following which, winter cooling driving convection and storms will result in mixing of the surface waters with cooler waters below. Within the subtropical gyres, this results in a thick layer of subtropical mode water (STMW) of near-homogeneous temperature and density. In the Atlantic, this is commonly referred to as 18° water, a phrase originally coined to describe the mode waters of the Sargasso Sea (Worthington, 1958). STMW subduct below the warmer surface mixed layer, separating it from the ocean thermocline below, as they circulate around the gyres. Near surface, widely spaced isotherms (constant temperature contours) or isopycnals (constant density contours) on ocean cross-sections indicate the presence of mode waters. Another water mass known for its high salt content is MOW (Mediterranean outflow water; Fig. 2.13 and see also Fig. 2.11). As

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freshwater in the inflowing water from the North Atlantic is removed by evaporation within the Mediterranean basins, its salinity increases as does its density. This Mediterranean water then overflows into the North Atlantic basin through the Straits of Gibraltar below the Atlantic inflow, sinking down the continental slope in a turbulent plume and entraining ambient fresher North Atlantic water until it settles at about 1000 m depth. At this depth, the water of MOW origin is observed as a prominent subsurface salinity maximum as it spreads northwest along the European continental slope (Reid, 1979). This salty MOW-origin water ultimately mixes with other fresher colder overflows of Arctic and Nordic seas origin as it spreads west, eventually mixing with water of Labrador Sea origin densified by very deep convection (Clarke and Gascard, 1983; Lilly et al., 1999). During this transit across the subpolar North Atlantic basins, these water masses mix with each other at depth eventually creating NADW. The high salt content of NADW allows it to retain more heat than most other water masses found at this depth. NADW is exported south as a deep western boundary current in the Atlantic, eventually reaching the Southern Ocean, where the ACC redistributes it into the Indian and Pacific Oceans. South of the ACC, winds drive upwelling of NADW to the surface where this water mass is known as circumpolar deep water (CDW). The very deep convection that has been observed in the Labrador and Nordic Seas reaches depths of around 1000 m as a result of intense winter cooling, but only over very localized areas 100 s of meters wide (Schott et al., 1994). On basin scales, winter convection in the subpolar seas deepens the mixed layers, with the deepening allowing seasonal shoaling of isopycnals (surfaces of constant density) within or immediately adjacent to the North Atlantic Current and the major boundary currents of the subpolar North Atlantic (McCartney and Talley, 1982). Within the deep mixed layers coinciding with the outcropping isopycnals, the properties of subpolar mode waters are set at temperatures in the 8–12 °C range (McCartney and Talley, 1982). Once formed, these subpolar mode waters spread through the upper ocean following the subpolar gyre circulation, with some portion being entrained into the deeper flows joining the deep western boundary current heading south. With the coldest air temperatures and the most extreme seasons, the polar regions are key regions of air-sea heat fluxes leading to water mass formation. In the seasonally sea-ice-covered oceans, there is a seasonal cycle in dense water formation associated with the freeze-up. Seawater loses heat to the atmosphere as it cools, increasing its density. As seawater begins to freeze, approximately 60% of its salt content is rejected and released to the water

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column below as dense, cold brines, with the remaining salt trapped in brine channels within the sea ice matrix that slowly continues to drain out over time (Lake and Lewis, 1970; Wettlaufer et al., 1997). These dense brines then sink through the water column, mixing with the ambient water until they reach a depth where they are neutrally buoyant. In coastal polynyas, areas where winds blown newly formed ice away from coastline, sea ice can continually form allowing the accumulation of the dense brines on the continental shelf below, creating high salinity shelf waters (Grumbine, 1991; Jacobs et al., 1979; Schauer, 1995). These high-salinity shelf waters are exported off the shelves, sinking as density currents down the continental slope. The sinking high-salinity shelf waters mix with and entrain fresher water masses forming bottom waters that fill up our abyssal oceans. The unintuitive thing about bottom waters is that, despite having origins in highsalinity shelf waters, they tend to be freshest water masses of their density class because of their extreme cold temperatures. Arctic intermediate and bottom waters circulate within the Arctic basins, and some portion is exported south into the Nordic Seas in the East Greenland Current (Rudels et al., 2005). This water is further transformed on its passage through the Greenland Sea until it spills over Greenland-Scotland ridge as the Denmark Strait overflow – the tallest waterfall on the planet, albeit underwater. Denmark Strait overflow water in turn eventually contributes to the formation of NADW. It is worth noting that as well as exported deep water, the Arctic also exports cold, very fresh, and therefore light, Polar Water south through both the Canadian Arctic Archipelago and Fram Strait. This light Polar Water has origins in the high Arctic river runoff; net precipitation and sea ice melt mixed with the saltier lower latitude Atlantic and Pacific Waters below. In the southern hemisphere, the high salinity shelf waters here are exported into the abyssal ocean as AABW (Foster and Carmack, 1976; Orsi et al., 1999). While both the Ross and Weddell Seas are known to be significant sources of high-salinity shelf waters, increasingly, it is thought that AABW is produced all along the Antarctic coastline, rather than in just a few main locations (Gordon, 2009; Williams et al., 2010). Like other water masses of Southern Ocean origin, AABW is spread into the Pacific, Indian, and Atlantic Oceans along the pathway of the ACC and is of particular interest in climate change studies because of the rapidity with which this water mass ventilates so much of our abyssal ocean. Dissolved atmospheric gases and particulate organic carbon entrained in AABW is swiftly transported from the surface to the abyss where it is stored for thousands of years.

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The other globally ubiquitous water mass formed in the Southern Ocean is Antarctic intermediate water (AAIW). AAIW is formed as a mode water on the northern edge of the ACC, where Ekman fluxes of fresh polar water across the ACC allow these waters to mix with waters of subtropical gyre character during winter in mixed layers that can easily become 400-m deep (Sloyan et al., 2010; Talley, 1996). Thus AAIW that forms is a very cold and comparatively fresh, eventually subducting below the STMW as it spreads northwards into the Pacific, Atlantic and Indian Oceans (Fig. 2.13). Temperature and salinity are not just useful in identifying specific water masses, they are also useful for inferring in what proportions two or more water masses have mixed to produce water of intermediate characteristics. For instance, any mixture C of two water masses A and B of given temperature and salinity characteristics will lie on the straight line drawn between the end-member temperature and salinities on a T-S diagram (Fig. 2.6, RHS). The relative proximity of C to either end-member A or B reflects the proportion of each end member in the mix, such that if the A-C distance is three times the C-B distance, then the mixture comprises one-fourth of A and three-fourths of B. Here we have discussed the main water masses that are found in most ocean basins; however, these water masses are subject to mixing and transformation with regional water masses in their passage through different ocean basins. Different ocean basins are subject to different forcings as a result of their geography and weather systems. For instance, the Mediterranean and, to a lesser degree, the Red Sea both contribute salt to the Atlantic and Indian Oceans, respectively, while the Pacific has no such source of salinity. Hence waters from each of these three basins are easily distinguished by their salt and freshwater content on T-S diagrams.

5. Oceanic impact on and sensitivity to Earth’s climate Geographical differences in water mass formation mechanisms enable the ocean to absorb and store solar radiation at lower latitudes while ventilating the ocean, losing heat to the atmosphere, and forming sea ice at high latitudes. Wind-driven ocean circulation patterns driving warm shallow western boundary currents polewards together with the sinking and equatorwards spread of dense cold bottom waters comprise the global overturning circulation (Fig. 2.14) of our oceans that redistributes heat within the Earth’s climate on a global scale. Within the global overturning circulation, water masses transport their properties from formation zones to other

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Fig. 2.14 Schematic of the overturning circulation of the global ocean with the warm water pathways depicted in red and cool dense water in blue.

locations where they are transformed into new water masses either by mixing with ambient waters, air-sea fluxes, or both. Overturning cells within the North Atlantic and Southern Ocean dominate this global thermohaline circulation, with weaker overturning in the Pacific and Indian Oceans completing the loop. In the North Atlantic, warm water carried in the Gulf Stream eventually feeds into the North Atlantic Current that enters the Iceland Sea before partially recirculating through the Irminger basin, while the remainder follows the European continental slope into the Nordic Seas and the Arctic Ocean. The Gulf Stream/North Atlantic Current forms the surface limb of the North Atlantic’s meridional overturning cell (AMOC, Johnson et al., 2019). The warm Gulf Stream water is freshened and cooled in the subpolar

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North Atlantic and Arctic basins, ultimately gaining density and being transformed into deep waters that participate in NADW formation. The southward flow of NADW forms the deep return limb of AMOC. In the Southern Ocean, instead of one overturning cell, there are two. Wind-driven upwelling in the ACC brings isopycnals associated with CDW to the surface at the interface of the two overturning cells. At the surface, some of the upwelled CDW mixes with the fresher polar mixed layer and is fluxed northwards across the ACC by Ekman transport, where it participates in AAIW formation. Together the northward Ekman transport and subducting AAIW form the uppermost limb of the Southern Ocean’s overturning cells. The farther south-outcropping CDW isopycnals are transformed by intense winter heat loss and mixing with dense waters of shelf origin to produce AABW that forms the deepest limb of the Southern Ocean’s overturning cell. The transport of ocean heat from low to high latitudes where it can warm the atmosphere makes the climate of these latitudes much milder than if the Earth had no ocean. This exchange of heat and freshwater (through evaporation and precipitation) between the oceans and Earth’s atmosphere effectively sets global climate and impacts weather patterns. On longer timescales, the meridional overturning circulation of the oceans supplies oceanic heat to the atmosphere at higher latitudes, making the high-latitude climates more productive and hospitable for life than on a planet without oceans. On shorter timescales of days to weeks, local oceanic hotspots can power storms that absorb both heat and draw water into them through evaporation, such that oceanic hotspots near land can power up tropical storms that wreak devastation on coastlines. The importance of ocean circulation in absorbing, storing, and redistributing heat around the planet cannot be overstated and is a direct result of differences in the density and specific heat capacity, cp, of seawater and the atmosphere. While scattering and absorption of downwelling solar radiation does occur within the atmosphere, it is largely transparent to visible light which easily passes through and reaches the surface of the planet. Seawater, on the other hand, absorbs visible light, storing this energy as heat, such that heat stored per m3 for a change in temperature, ΔT, is given as ΔQ ¼ ρscpsΔT, where the s subscript denotes seawater. The specific heat capacity of seawater cps ¼ 3900 J kg1 K1 and seawater density is typically 1026 kg m3, while the density and specific heat capacity of air at 300 K are nominally 1 kg m3 and cpa ¼ 1000 J kg1, respectively. Effectively, the same heat input will change surface air temperature by approximately 4000 times the

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temperature change in seawater. This means that as increasing greenhouse gases have trapped more longwave radiation, the oceans have been critical for keeping the planet cool, as measurements and historical reconstructions show that the oceans have absorbed about 90% of the excess heat (Cheng et al., 2020; Zanna et al., 2019). Yet this increased ocean heat storage has already had an impact on life on the planet. First, the oceans have expanded as they warmed, contributing to sea level rise (Domingues et al., 2008; Levitus et al., 2012). Sea level rise has accelerated coastal erosion, threatened salt marshes, will reconfigure barrier islands (FitzGerald et al., 2008; Neumann et al., 2015), and is inundating low-lying islands at a rate that threatens the medium-term existence of several island nations. Second, warmer ocean temperatures have resulted in an increasing frequency of coral bleaching events (Glynn, 1991; van Hooidonk et al., 2016; Hughes et al., 2017) that purges corals of their symbiotic zooxanthellae, imperiling the survival of many reefs and changing their ecosystems (Hughes et al., 2018). Third, rising temperatures reduce the solubility of gasses in the ocean, including oxygen which is critical for respiration. Indeed, the reduction in oxygen concentration levels in many regions of the ocean has already been documented (Breitburg et al., 2018; Shaffer et al., 2009), with numerical models predicting future dire consequences for phytoplankton that are the primary producers of the ocean (Sekerci and Petrovskii, 2015; see also Chapter 13). Any decrease in ocean primary production, in turn, reduces the ocean’s ability to fix atmospheric carbon dioxide and sequester it from the atmosphere and, in other words, reducing the ocean’s role as a critical store for atmospheric carbon dioxide. Finally, the warming ocean’s impact on Arctic sea ice has been increasing in the last few decades (Barton et al., 2018; Lind et al., 2018; Polyakov et al., 2020), contributing to the loss in seasonal sea ice and reducing the earth’s albedo with consequences for marine ecosystems and the planetary energy budget. Oceanic warmth is also undermining tidewater glaciers in Greenland (Millan et al., 2018; Rignot et al., 2016; Wood et al., 2018) and ice shelves in Antarctica (DeConto and Pollard, 2016; Jacobs et al., 1979; Rignot et al., 2013; Stanton et al., 2013). The consequence of this melting is to accelerate the glacier streams toward the ocean, enhancing the contribution of land-based ice to sea level rise. Clearly, on multiple timescales from days to decades, how the ocean transports heat, freshwater, and other tracers around the globe has profound consequences for our planet. And although proper energetics of ocean circulation is still an ongoing discussion, it is evident that tides and the energy

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they dissipate drive ocean mixing and mediates the transfer of heat between oceanic layers, and the atmosphere, in different ways that impacts the global overturning circulation both today and throughout Earth’s history (Green et al., 2009; Schmittner et al., 2015; Wilmes et al., 2021). Thus no story of our oceans would be complete without a focus on tides.

Acknowledgments The authors acknowledge the following funding for this contribution: YDL, the UK-German NERC-BMBF Changing Arctic Ocean PEANUTS project (NE/ R01275X/1), and NERC ArctiCONNECT consortium NE/V005855/1. The ARMOR3D hydrographic product can be found at https://doi.org/10.48670/moi00052. EN.4.2.2 data were obtained from https://www.metoffice.gov.uk/hadobs/en4/ and are © British Crown Copyright, Met Office, provided under a Non-Commercial Government License http://www.nationalarchives.gov.uk/doc/non-commercial-governmentlicence/version/2/.

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CHAPTER 3

A brief introduction to tectonics João C. Duarte

Geology Department and Instituto Dom Luiz (IDL), Faculty of Sciences of the University of Lisbon, Lisbon, Portugal

1. Tectonics Tectonics is the study of lithospheric motion and deformation on actively convecting silicate planetary bodies (Stern, 2018). Most planets and moons have some kind of tectonics. On Earth, the brittle outer layer is broken up into several large lithospheric plates interconnected by a network of plate boundaries (Fig. 3.1). This type of tectonics is known as plate tectonics. Many fundamental geological processes such as mountain building, earthquakes, and volcanism occur along the plates’ boundaries (Duarte and Schellart, 2016). The lithospheric plates are rigid and move in relation to each other over a more ductile layer – the asthenosphere (Fig. 3.2). The movement at the surface expresses the planet’s internal dynamics and provides the substrate for other complex emergent processes and feedbacks to occur, such as the supercontinent cycle, the supertidal cycle, and the diversification of life itself. In this chapter, I will first summarize the ideas that lead to the realization of plate tectonics and then introduce some of the main concepts and processes that will be of value to this book.

1.1 Early ideas Like all great ideas, it is difficult to tell when it all started. The ancient Greeks proposed diverse explanations for the seismological and volcanic processes observed around the Mediterranean, which were later revived during the Renaissance period. These include the vertical motions of the Earth’s surface proposed to occur as an explanation for the existence of marine fossils inside the continents, sometimes on the top of mountains (Galopim de Carvalho, 2014). Notwithstanding, it was only during the production of the first maps of the Atlantic Ocean, in the 16th century, that the cartographer Abraham Ortelius noted its margins fitted like a puzzle and suggested that they were once together, and later separated by some sort of cataclysm.

A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00003-0

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Fig. 3.1 Plates and plate boundaries (Duarte and Schellart, 2016). (Modified from Schellart, W. P., Stegman, D.R., Farrington, R.J., Moresi, L., 2011. Influence of lateral slab edge distance on plate velocity, trench velocity and subduction partitioning. J. Geophys. Res. 116, B10408. https://doi.org/ 10.1029/2011JB008535; courtesy of Wouter Schellart). © 2016 American Geophysical Union, by John Wiley & Sons, Inc.)

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Fig. 3.2 Simplified 2D cross section of the Earth showing the interior of the Earth. Note that subducting slabs are zones of downwelling and plumes represent zones of upwelling.

Antonio Snider-Pellegrini, in 1858, based on both geometric arguments and the presence of similar fossils on separate continents, drew a “supercontinent” in which the Americas were joined to Africa and Eurasia (Snider-Pellegrini, 1858; Fig. 3.3). Snider-Pellegrini lacked a proper scientific explanation but proposed that the New World separated from the Old World during the biblical flood. Over the subsequent years, several

Fig. 3.3 Snider-Pellegrini’s illustration of the closed and opened Atlantic Ocean (1858). (From Snider-Pellegrini, A. (1858). La Creation et ses mystères devoiles (Creation and its mysteries revealed). Public domain.)

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arguments were put together in favor of continental movements. However, most scientists were fixists, and alternative explanations such as the existence of past land bridges and/or an expanding (or shrinking) Earth were still the most popular explanations (Galopim de Carvalho, 2014). During the turn of the 20th century, Alfred Wegener become obsessed with this problem and collected a series of evidence to sustain his mobilist ideas. He presented the concept of continental drift in 1912, at the annual meeting of the German Geological Society, and compiled evidence in a book published in 1915, titled “The Origin of Continents and Oceans” (Wegener, 1929). While this was the keystone for the basis of a new theory of the Earth, Wegener always recognized that it did not come as an epiphany. He acknowledged that much of his work was built on the shoulders of some of his contemporaries, namely, Franklin Coxworthy, Roberto Mantovani, William Pickering, and Frank Taylor (Wikipedia, 2022). Curiously, the latter proposed that the continents have crept away from an existent supercontinent because of the higher tidal forces in the Cretaceous (Powell, 2015). Others that contributed to these mobilist ideas were James Dana, Charles Lyell, and Eduard Suess. Suess, for example, proposed the existence of a past supercontinent, which he named Gondwana, and of a large ocean that once existed between Africa and Europe, the Tethys (Wikipedia, 2022). The continental drift hypothesis states that all the continents were once gathered in a supercontinent, Pangea, and drifted away over geological time scales. It is important, however, to note two aspects. This hypothesis does not say much about what happened before Pangea, and, more importantly, the continents moved over a relatively stagnant oceanic seafloor, like ships moving through the water. This had important consequences that led the theory to be rejected for almost 50 years. A proper physical mechanism was lacking, and rock rheology was still in its infancy. How could solid rock flow over solid rock?

1.2 Paradigm shift A tentative explanation for the mechanisms behind continental drift came a few years later by the hand of Arthur Holmes, who, in 1931, proposed that the cause of surface tectonics on Earth was mantle convection, resulting from radiogenic heating (Holmes, 1931). According to his view, there were zones of upwelling that would stretch the crust and cause the continents to separate, and zones of downwelling, over which there was shortening and

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the formation of mountains. This is close to our present view of the Earth, but there was a shortcoming: these upwelling zones did not reach the surface. The surface lid was deformable but not broken. There were no plates. Shortly after WWII, using new radar and echo sounding technology, extensive mapping expeditions were carried out over the seafloor of the world. Maurice Ewing’s geophysical team was key in converting this data to readable maps (mostly done by Marie Tharp and Bruce Heezen). Ewing and Heezen’s (1956) work showed “great rift systems” running through most oceans, which revealed ocean floor was not one continuous sheet. This partly validated Holmes’ convection theory, but the picture was still not yet complete. It took a few more years, until the early 1960s, for Harry Hess and Robert Dietz to introduce the idea of seafloor spreading, later confirmed by Lawrence Morley, Fred Vine, and Drummond Matthews, based on the identification of alternating magnetic stripes on the seafloor (Hess, 1962; Morley and Larochelle, 1964; Dietz, 1961; Vine and Matthews, 1963). Hess (1962) proposed that the Atlantic Ocean was growing as the result of seafloor spreading, at the same time that the Pacific was closing as the result of consumption of ocean floor at subduction zones along its margins.

1.3 The theory of plate tectonics It all came together in the second half of the 1960s. For that to happen, Tuzo Wilson had to come up with the last piece of the puzzle: transform faults. While long fracture zones were already known to exist on the seafloor, it was Wilson who understood what they were and how they functioned (Wilson, 1965). And by doing so, he could connect spreading centers to subduction zones, which when combined defined a network of linear features that separated different lithospheric plates (see Fig. 3.1). Immediately after this discovery, several models dividing the surface of the Earth into lithospheric plates were proposed by Jason Morgan and Xavier Le Pichon (Le Pichon, 1968; Morgan, 1968). More or less simultaneously, Dan Mackenzie and Robert Parker developed a kinematic model (McKenzie and Parker, 1967) of what then become known as “the new global tectonics” and later “the theory of plate tectonics.” Suddenly, it all made sense. Around this time, Bryan Isacks, Jack Oliver, and Lynn Sykes understood that the global distribution of earthquakes aligned along these plate boundaries, while the interior of the plates was

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relatively free of earthquakes (Isacks et al., 1968). Plate boundaries also coincide with the location of volcanoes, mountain chains, sedimentary basins, anomalous geothermal gradients, the formation of mineral deposits, and many other geological phenomena and processes. Plate tectonics unified many disciplines of solid Earth sciences, including tectonics, structural geology, mineralogy, petrology, volcanology, seismology, paleontology, stratigraphy, and sedimentology. This was a scientific revolution, and plate tectonics is now regarded as the unified theory of the solid Earth.

1.4 The modern conception of plate tectonics Plate tectonics describes that the surface of the Earth is divided into several rigid plates that move in relation to each other over the more ductile asthenosphere. The plates can diverge, converge, or move laterally (Fig. 3.4). Divergent plate boundaries are places where new plate material is created. These are volcanic regions with regular, moderate seismicity. In convergent zones, plates can either collide, forming mountain chains, or one can plunge below the other in subduction zones, without forming much topographic elevation. Convergent zones are known for their irregular, high-magnitude seismicity, associated with deformation and metamorphism, and significant volcanic activity, often expressed by a volcanic arc. Transform plate boundaries link the other two types of boundaries, for example, “transforming” the motion of a divergent plate boundary into the motion of a convergent boundary. These are regions with low volcanic activity and low-to-moderate seismicity, with rare high-magnitude events (see Duarte and Schellart, 2016). One thing that is important to note is that in plate tectonics we refer to the lithosphere and lithospheric plates. This is not to be confounded with crust. The terms crust, mantle, and core refer to layering based on Earth’s chemical composition. The crust is made of light elements, such as oxygen, silicon, and aluminum. The mantle is, instead, composed of heavier iron and magnesium silicates, whereas the core is made of metallic iron and nickel. Notwithstanding, the Earth can also be divided according to its mechanical layering, which expresses how rocks respond (mechanically) to temperature and pressure at different depths. The lithosphere is the upper and mechanically more rigid layer of the Earth. Its thickness varies between 10 and 100 km in the oceans, reaching up to 200 km, or more, under the continents. The lithosphere is made of the crust and the upper part of the upper mantle. Below the lithosphere,

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Fig. 3.4 Types of plate boundaries: convergent, divergent, and transform (Duarte and Schellart, 2016). (© 2016 American Geophysical Union, by John Wiley & Sons, Inc.)

there is a more ductile layer, the asthenosphere, that extends down to 400–660 km (the base of the upper mantle). From 660 km down to the core-mantle boundary (at 2900 km), we find the lower mantle, which is thought to be 2 orders of magnitude more viscous than the upper mantle (Turcotte and Oxburgh, 1972; Karato, 2010). While initially plate tectonics was a purely kinematic theory that described the motion of tectonic plates, geologists soon questioned what forces were driving it. The first suspect was a revival of the continental drift

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hypothesis: mantle convection (Turcotte and Oxburgh, 1972). However, it was soon realized that plates could not be passively driven by underlying convection currents. Instead, the plates seemed to themselves be part of the convective system (Turcotte and Oxburgh, 1972). According to this view, now well supported by observations and modeling, tectonic plates are not simply the superficial expression of mantle flow. They are an integral part of mantle convection and one of the main drivers of mantle flow (at least the oceanic lithosphere; the continents are a different story as they tend to remain at the surface for buoyancy reasons). As my colleague Wouter Schellart once told me: “plate tectonics is mantle convection.” The oceanic lithosphere is the cold top thermal boundary layer of the mantle and subduction zones represent zones of downwelling, while deep plumes generated at the core-mantle boundary are zones of upwelling (see e.g., Torsvik et al., 2016). We now know that a significant part of a plate’s motion and mantle flow results from the sinking of lithospheric slabs at subduction zones, which can reach the core-mantle boundary. This is confirmed by a plate’s kinematics which show that most plates move in the direction of subduction zones and that plates that are attached to subducting slabs move almost 1 order of magnitude faster than plates that are not subducting (see Fig. 3.1). One consequence is that spreading centers are mostly passive, which implies a plate’s divergent motion is a consequence rather than a cause of their movement. The drivers of plate tectonics can be divided into three main forces (Forsyth and Uyeda, 1975): 1) The slab-pull force: the force imparted to the plates by the sinking of lithospheric slabs at subduction zones. This force is almost 1 order of magnitude higher than the others and is the main contributor to the plate’s motion. 2) The ridge-push force: this force results from the elevation of ridges with respect to the adjacent abyssal plains. The ridges sit higher than the surrounding abyssal plains because seafloor spreading leads to local upwelling and elevation of the asthenosphere, which causes the lithosphere to glide down away from the spreading centers. 3) The slab-suction force: this is the force responsible for keeping the plates together when one plunges below the other. This plunge causes trenches to migrate, and when that happens, the overriding plate follows the subducted plate by suction. There is a fourth force, sometimes referred to as plume push force, which results from the arrival and drag of an impinging mantle plume (e.g., Cande and Stegman, 2011). This force acts locally and is more ephemeral.

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2. Earth’s tectonic cycles The Earth’s lithosphere is partially composed of oceanic or continental crust, and both often coexist on the same plate. However, because the oceanic lithosphere is negatively buoyant and continental lithosphere is positively buoyant in relation to the underlying asthenosphere (Cloos, 1993), only the oceanic portions of plates subduct and are involved in deep mantle convection. Their continental counterparts, while being dragged by the sinking of the oceanic segments, resist being subducted at subduction zones. Continents, therefore, have mostly remained at the surface of the Earth since they formed. They are like rafts that are not involved in deep mantle convection. This leads to interesting effects, such as the Wilson cycle and the grand supercontinental cycle (see, e.g., Dewey and Spall, 1975 for the Wilson Cycle; and Nance et al., 1988; Yoshida and Santosh, 2017 for the supercontinent cycle; Davies et al., 2018 presents an overview).

2.1 The Wilson cycle The Wilson cycle, named in honor of Tuzo Wilson, can be better understood using the example of the Atlantic Ocean, the ocean to which the concept was first applied (Wilson, 1966; Dewey, 1969). When the supercontinent Pangaea broke up around 180 Ma, it formed an internal pristine ocean – the Atlantic – made of new oceanic lithosphere that grew from a newly formed spreading center. New ocean floor is hot and buoyant and floats over the underlying asthenosphere. However, as the new crust forms, the lithosphere progressively moves away from the spreading center, cools, and densifies (Cloos, 1993). About 10–20 million years after its formation, the oceanic lithosphere becomes negatively buoyant and sags, forming deep abyssal plains. At this point, the only thing preventing the oceanic lithosphere from sinking further into the upper mantle is the fact that it is attached to buoyant continents, on one side, and buoyant spreading centers, on the other – like two persons holding a sheet between them. However, this metastable equilibrium only lasts for a limited amount of time. For reasons that we still do not fully understand, the oceanic lithosphere eventually founders and starts to subduct. The timings are quite striking: almost all the oceanic lithosphere on Earth is younger than 200 million years (Muller et al., 2008). Note that the Earth is much older – 4600 million years – and that oceanic lithosphere has been continuously forming for at least one billion years (Bradley, 2011). This suggests that the Atlantic seafloor is already old and must be soon consumed in subduction zones. In fact, it already is, in two

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narrow subduction zones: the Lesser Antilles Arc and the Scotia Arc, which limit to the east the Scotia and the Caribbean plates, respectively (see Fig. 3.1; Duarte et al., 2013, 2018). These subduction zones are expected to extend to become ocean wide and consume the Atlantic seafloor, pulling the continents back together. This means that the Atlantic Ocean will soon close (geologically speaking), very much like Pangaea in reverse. This is the simplest form of a Wilson Cycle: an ocean opens, grows, and then shrinks and closes. In a way, it is like the oceans have a life cycle, and that is the reason the Wilson Cycle is also known as the oceanic cycle (Dewey, 1969). However, on Earth, several oceans coexist. This complicates things further but makes everything even more interesting.

2.2 The supercontinent cycle On Earth, several oceanic basins and continental blocks coexist at a given time. These oceanic basins all have their life cycles. When a supercontinent breaks up, it will fragment into a number of continents separated by pristine internal oceans. These oceans will eventually close, each corresponding to the termination of a Wilson cycle. If several oceans close simultaneously, leading to the gathering of all (or almost all) continental masses, a new supercontinent forms (see, e.g., Davies et al., 2018). The period that spans the tenure of one supercontinent on to the formation of the next one is referred to as the supercontinent cycle. A corollary of this is that Wilson cycles are, in general, of a lower order than supercontinent cycles. There may be several Wilson cycles (one for each ocean that closes) within a higher-order supercontinent cycle (see Duarte et al., 2018; Davies et al., 2018). Geologists have good evidence that there were at least two major supercontinents on Earth, Rodinia (formed at 1 Ga), and Pangea (formed at 300 Ma), defining a complete supercontinent cycle (Rogers and Santosh, 2003; Meert, 2014). Some authors also consider that a short-lived supercontinent, Pannotia, might have existed between Rodinia and Pangea, at 660 Ma (Nance and Murphy, 2018). An alternative is to consider that Pannotia was a megacontinent, in agreement with the idea that the formation of a supercontinent is usually preceded by the formation of one or two smaller megacontinents (Wang et al., 2021). There is also some evidence that supercontinents existed before Rodinia, namely, Columbia (at 1.7 Ga), Kenorland (at 2.5 Ga), and the more speculative Ur (at 3 Ga) and Vaalbara (at 3.5 Ga) (See Davies et al., 2018, for a discussion). However, we do not really know when present-day-like plate tectonics started (Stern, 2018; Palin

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and Santosh, 2021); our best estimates point to sometimes between 1 and 2 Ga. When we discuss Earth before that, the Wilson and supercontinent cycles are fuzzy concepts. Furthermore, geologists think that we are presently halfway through a supercontinent cycle, and that a new supercontinent will form in the future (e.g., Davies et al., 2018; Yoshida, 2016). But when, and what it will look like? Because the Earth’s surface is finite, the continental blocks will eventually collide at some point, and this can happen in a more or less synchronized way. Geodynamicists still debate if there is a dynamic reason for that, for example, after a dispersal phase, the continents are pulled toward mantle downwellings, or if it is simply a random process constrained by the size ratio between the area of the continents and the surface area of the Earth (see Duarte et al., 2018). Whatever the reason, there seems to be some kind of statistical cyclicity in the timings of the supercontinent cycle, with periods of around 400–600 Myrs (e.g., Bradley, 2011), suggesting that the next supercontinent may form in the next 200–300 Myrs. But where will it form? When a supercontinent breaks up, several continental blocks are sent around the globe in different directions. The motions are dependent on where the new rifts form and on the configuration of the subduction zones (downwellings) surrounding the supercontinent (see, e.g., Yoshida and Santosh, 2017). There is, therefore, some randomness in the initial conditions of each supercontinent cycle. This leads to interesting effects and suggests that the supercontinents can form in different ways. We sometimes refer to this as modes of supercontinent formation, and they can be understood as a sort of end members. These include introversion, extroversion, orthoversion, and combination (Murphy and Nance, 2003; see Fig. 3.5; Davies et al., 2018). In an introversion scenario, a supercontinent breaks up and internal oceans form, very much like the break-up of Pangaea led to the formation of the Atlantic. Then, after a period of dispersal, new subduction zones form inside these internal oceans, leading to their closing. This will bring the continents back together to form the next supercontinent. Here, the supercontinent cycle and the Wilson cycles of the internal oceans are in phase, i.e., the closing of the oceans that formed as a consequence of the break-up of the previous supercontinent are the ones that will close to form the following supercontinent. This scenario will happen on the future Earth if the Atlantic closes to form the next supercontinent, in a scenario that leads to the formation of a supercontinent called Pangea Ultima (see Fig. 3.5; Scotese, 2003; Davies et al., 2018).

Fig. 3.5 Four proposed future supercontinents: Pangea Ultima, Novopangea, Aurica, and Amasia. Adapted from Davies, H., Green, M., Duarte, J.C., 2018. Back to the future: testing different scenarios for the next supercontinent gathering. Global Planet. Change 169, 133–144. https://doi.org/10.1016/j.gloplacha.2018.07.015.

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An opposite picture happens in an extroversion scenario. Here, after the break-up of the supercontinent, the continents travel around the Earth until they meet in the antipodes, closing what was once an external ocean. Note that in this case, the supercontinent cycle and the Wilson cycles of both the newly formed internal ocean and the external ocean are out of phase. This would correspond to the closing of the Pacific Ocean in the ongoing supercontinent cycle, leaving the Atlantic open. The resulting hypothetical future supercontinent was named Novopangea by Nield (2007). Orthoversion is different to the previous two scenarios. In this scenario, the continents move away to rejoin approximately 90° away from the former supercontinent. There are some dynamic reasons for this, which are related to how the mantle reorganizes as a consequence of the closing of the preceding supercontinent. This scenario would correspond to the present continents gathering around the North Pole, except Antarctica which stays where it is. This hypothetical supercontinent was named Amasia (Mitchell et al., 2012). The previous scenarios assume that there are only two main continental blocks moving around the Earth separated by two oceans. This is likely a bias imparted by what we see on Earth today, with Africa and Eurasia on one side, and the Americas on the other. This often results in a slippery question: will the Atlantic or the Pacific close to form the next supercontinent? Note that, if there is more than one ocean on Earth, this question loses its strict meaning. If there is a third ocean, let’s say the Indian Ocean, then combination scenarios are possible. This is precisely the scenario of a future supercontinent named Aurica, in which both the Atlantic and the Pacific close simultaneously, by introversion and extroversion, respectively (Duarte et al., 2018). There are dynamic reasons to consider that these (combination) scenarios are more likely, in particular, the fact that the oceanic lithosphere seems to not be able to survive at the Earth surface for long, and that, therefore, the “old” Atlantic and Pacific oceans should close sometime soon. There is also evidence that Rodinia and Pangaea were formed by a combination of introversion and extroversion, with a component of orthoversion.

2.3 The supertidal cycle One question that might have crossed the mind of the reader at this point is why is tectonics important for the study of tides? Well, as continents move around the Earth surface, they exert a fundamental control on the evolution of the ocean basins. Tectonics sets the geometry, the morphology, the depth,

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and the size of the oceans, which, at geologic time scales, are constantly changing. This means that ocean dynamics, including tidal dynamics, is constantly changing as well. It is obvious when we look at observed tides that they don’t fit the theoretical picture (see Chapter 4, Gerkema, 2019, or Pugh and Woodworth, 2014 for details). This is partly because of the influence of continents (see also Blackledge et al., 2020), which force the tide to propagate along them, and which can set up tidal resonances. The Atlantic presently has large semidiurnal tides due to a resonance related to its width (e.g., Platzman, 1975; Arbic and Garrett, 2010; Green, 2010). This resonance did not exist just after the Pangaea supercontinent broke up and will likely fade away in the near geological future (Green et al., 2018 and later chapters in this book). Furthermore, if one day the Pacific closes, it will also pass through a similar period of semidiurnal resonance, when its width narrows to that of the present-day Atlantic (e.g., Davies et al., 2020). This illustrates how these basin-wide tidal resonances are intimately linked to the Wilson cycle, and how during a full supercontinent cycle, tides may go through oscillating periods of low and high energy. This cyclicity in tidal energy associated with the supercontinent cycle is known as the supertidal cycle (Green et al., 2018; Davies et al., 2020) and will be explored further in Chapters 6–10.

References Arbic, B.K., Garrett, C., 2010. A coupled oscillator model of shelf and ocean tides. Cont. Shelf Res. 30, 564–574. Blackledge, B.W., Green, J.A.M., Way, M.J., Barnes, R., 2020. Tides on other Earths: implications for exoplanet and palaeo-tidal simulations. Geophys. Res. Lett. 47, e2019GL085746. Bradley, D.C., 2011. Secular trends in the geologic record and the supercontinent cycle. Earth Sci. Rev. 108, 16–33. https://doi.org/10.1016/j.earscirev.2011.05.003. Cande, S.C., Stegman, D.R., 2011. Indian and African plate motions driven by the push force of the Reunion plume head. Nature 475, 47–52. https://doi.org/10.1038/ nature10174. Cloos, M., 1993. Lithospheric buoyancy and collisional orogenesis: subduction of oceanicplateaus, continental margins, island arcs, spreading ridges, and seamounts. Geol. Soc. Am. Bull. 105, 715–737. https://doi.org/10.1130/0016-7606(1993). Davies, H., Green, M., Duarte, J.C., 2018. Back to the future: testing different scenarios for the next supercontinent gathering. Global Planet. Change 169, 133–144. https://doi. org/10.1016/j.gloplacha.2018.07.015. Davies, H.S., Mattias Green, J.A., Duarte, J.C., 2020. Back to the future II: tidal evolution of four supercontinent scenarios. Earth Syst. Dynam. 11, 291–299. https://doi.org/ 10.5194/esd-11-291-2020. Dewey, J.F., 1969. Continental margins: a model for conversion of Atlantic type to andean type. Earth Planet. Sci. Lett. 6, 189–197. https://doi.org/10.1016/0012-821X(69)90089-2.

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Murphy, J.B., Nance, R.D., 2003. Do supercontinents introvert or extrovert? Sm-Nd isotopic evidence. Geology 31, 873–876. https://doi.org/10.1130/G19668.1. Nance, R.D., Murphy, J.B., 2018. Supercontinents and the case for Pannotia. Geol. Soc. Lond. Spec. Publ. 470, 65–86. https://doi.org/10.1144/SP470.5. Nance, R.D., Worsley, T.R., Moody, J.B., 1988. The supercontinent cycle. Sci. Am. 259, 72–79. Nield, T., 2007. Supercontinent. Granta Books, London, p. 287. Palin, R.M., Santosh, M., 2021. Plate tectonics: what, where, why, and when? Gondw. Res. 100. https://doi.org/10.1016/j.gr.2020.11.001. Platzman, G.W., 1975. Normal modes of the Atlantic and Indian Oceans. J. Phys. Oceanogr. 5, 201–221. Powell, J.L., 2015. Four Revolutions in the Earth Sciences: From Heresy to Truth. Columbia University Press. Pugh and Woodworth, 2014. Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes. Cambridge University Press, ISBN: 9781139898706. Rogers, J.J.W., Santosh, M., 2003. Supercontinents in earth history. Gondw. Res. 6, 357–368. https://doi.org/10.1016/S1342-937X(05)70993-X. Scotese, C.R., 2003. Palaeomap Project. http://www.scotese.com/earth.htms. Snider-Pellegrini, A., 1858. La creation et ses myste`res devoiles ; ouvrage ou` l’on expose clairement la nature de tous les etres, les elements dont ils sont composes et leurs rapports avec le globe et les astres, la nature et la situation du feu du soleil, l’origine de l’Amerique, et de ses habitants primitifs, la formation forcee de nouvelles plane`tes, l’origine des langues et les causes de la variete des physionomies, le compte courant de l’homme avec la terre, etc. Avec dix gravures. Paris, A. Franck. Stern, R.J., 2018. The evolution of plate tectonics. Philos. Trans. Royal Soc. A: Math. Phys. Eng. Sci. https://doi.org/10.1098/rsta.2017.0406. Torsvik, T.H., Steinberger, B., Ashwal, L.D., Doubrovine, P.V., Tronnes, R.G., 2016. Earth evolution and dynamics—a tribute to Kevin burke. Can. J. Earth Sci. 53 (11), 1073–1087. https://doi.org/10.1139/cjes-2015-0228. Turcotte, D., Oxburgh, E., 1972. Mantle convection and the new global tectonics. Annu. Rev. Fluid Mech. 4, 33–66. Vine, F.J., Matthews, D.H., 1963. Magnetic anomalies over oceanic ridges. Nature 199, 947–949. https://doi.org/10.1038/199947a0. Wang, C., Mitchell, R.N., Murphy, J.B., Peng, P., Spencer, C.J., 2021. The role of megacontinents in the supercontinent cycle. Geology 49, 402–406. Wegener, A., 1929. Die Entstehung der Kontinente und Ozeane, forth ed. Friedrich Vieweg & Sohn, Braunschweig. Wikipedia (2022) https://en.wikipedia.org/wiki/Continental_drift. Wilson, J.T., 1965. A new class of faults and their bearing on continental drift. Nature 207, 343–347. https://doi.org/10.1038/207343a0. Wilson, J.T., 1966. Did the Atlantic close and then reopen? Nature 211, 676. Yoshida, M., 2016. Formation of a future supercontinent through plate motion—driven flow coupled with mantle downwelling flow. Geol. Soc. Am. 44, 755–758. https://doi.org/ 10.1130/G38025.1. Yoshida, M., Santosh, M., 2017. Geoscience Frontiers voyage of the Indian subcontinent since Pangea breakup and driving force of supercontinent cycles: insights on dynamics from numerical modeling. Geosci. Front., 1–14. https://doi.org/10.1016/j. gsf.2017.09.001.

CHAPTER 4

Why is there a tide? Sophie Ward, David Bowers, Mattias Green, and Sophie-Berenice Wilmes

School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom

1. Introduction to tides For millenia, people have watched the ocean from the seashore and asked the question: "Why is there a tide?" It was apparent to early civilizations that there was a link between the tide and the Moon, but how exactly did this link work and why were there two tides each day when the Moon passed overhead just once? The answer had to wait until the publication, in Latin, of Isaac Newton’s Principia in 1687. Newton proposed a solar system in which the Sun, planets, and moons attracted each other with the force of gravity that varied inversely as the square of their separation. It was the variation in gravity that was the key: it created the stretching forces that make tides. At a stroke, this brilliant piece of thinking explained the origin of the tides and the principal rhythms of the tide in the ocean, but there was still work to be done to understand the way that the ocean responds to the various forces exerted on it.

1.1 The importance of tides Ocean tides, characterized by the periodic rise and fall of sea level, are the longest waves that propagate through the oceans. These oscillations of ocean waters occur under the influence of the combined gravitational forces of the Moon and the Sun, which exert a gravitational pull on the Earth’s water bodies. The timing of the tide (tidal phase) is dependent on its location, whereas the height of the tide (tidal amplitude) is greatly influenced by the water depth and shape of ocean basins. Many eminent scientists and mathematicians of the last 400 years (including Galileo, Newton, Euler, Bernoulli, Laplace, Jeffreys, and Munk) have worked to understand, calculate, and predict the tides. Indeed, some of the earliest computers were developed to predict the tides. The tides in the real world are complex – a long-period

A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00001-7

Copyright © 2023 Elsevier Inc. All rights reserved.

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sloshing of ocean waters in a rotating system, within ocean basins and shelf seas of varying depths, with submarine ridges and mountains, and funneling channels and estuaries. Tides are fundamental to several natural systems on Earth: from influencing large-scale dynamical systems to playing a vital role in small-scale marine ecosystems processes. Along the coastline, unique marine ecosystems are found in the intertidal zone, the area of shoreline between high and low tides, which is alternately wet and dry. Plants and animals living in this zone must be able to survive in the air as well as underwater; they are exposed to harsh sunlight and often must withstand crashing waves. Ocean tides affect humans in our quest for food and shelter (and exploration!). The tide determine when we can fish or forage for food in the intertidal region; can help or hinder marine navigation; interacts with the built environment where we have developed the coastline; and the tide can even influence weather (e.g., Arnold et al., 2021). The tide is an important consideration in coastal developments and in the planning of coastal defense structures. It has also been shown that the phase of the tide can modify the propagation of storm surges (Horsburgh and Wilson, 2007) and can amplify the impacts of storms on the coastline (Rulent et al., 2021). The tide has the potential to contribute to the decarbonization of global energy sources. Electricity can be generated from the tide, by extracting either kinetic energy from tidal currents or by tapping into the potential energy of the vertical tidal motion by holding the tide behind a dam or barrage. Tidal energy, although in relative infancy in comparison to other sources of renewable energy (e.g., wind or solar energy), has the distinct advantage of being a predictable source of energy generation owing to the repeating nature of the tide – it is not reliant on either the wind blowing or the Sun shining, for example. The tide causes the most predictable – and one of the most highfrequency – changes in sea level. Although this rise and fall in sea level is most apparent in the coastal zone, the tide also influence important processes further offshore. For example, in seasonally stratified waters of the temperate and polar shelf seas, the tide can influence the position of the tidal mixing fronts, which delineate seasonally stratified and generally well-mixed waters. These mixing fronts are important features of many shelf seas as they are biologically highly productive and biogeochemically active. The mixing, driven by tidal stirring, and the stratification, resulting from solar heating of the water column, are in balance at the position of the tidal mixing front (Simpson and Hunter, 1974). The positions of the fronts vary with the spring-neap cycle, as well as seasonally (Simpson and Bowers, 1981). Further

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still, the distribution of seasonally stratified and fully mixed waters influences atmospheric CO2 concentrations, through the export of dissolved inorganic carbon into the deep ocean (Tsunogai et al., 2016). While this is not an exhaustive list of why the tide is important, it is important to note here that perhaps the most physically far-reaching influence of the tide, long-term, is on the change in day length. The tide is responsible for slowing down the spin of the Earth, continuously increasing the length of our day and fundamentally impacting on the evolution of the Earth-Moon system itself (see Chapter 17 for a discussion). Here, we introduce the fundamentals of tides, tidal theory, and describe the tide in the real world.

1.2 The ups and downs of the seas The tide is the regular rise and fall of sea level induced by the gravitational forces from the Sun and Moon. In most places on Earth, there are two high waters and two low waters in a day (see Fig. 4.1). As outlined in Chapter 1, it has long been realised that the tide is linked to the Moon: high tide always occurs a certain time after the Moon rises in the night-time sky and the tides are larger around full Moon and new Moon. But why are there two high

Fig. 4.1 Tide gauge data. Examples of tidal signals from Newport, San Francisco, Adelaide, and Scott base. Data from GESLA, see gesla.org for details.

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tides per day? And why is the tidal signal in Newport (UK) so different to that in San Francisco (USA) or Adelaide (Australia)? And what is happening to the tides around Antarctica at Scott Base in Fig. 4.1D? We will look at the forces that cause the tide and how these change in time and space. But first, let’s think about how the tide propagates on the Earth. There are many different types of waves in the ocean, including surface waves generated by wind and earthquake-triggered tsunamis. Although we usually think of the short period waves that we witness on the coast, the tide is another wave (a very long period wave). At about 1000 times wider than it is deep, the ocean is (perhaps surprisingly) a relatively shallow body of water as far as tide waves are concerned (at least it can be treated as such mathematically). Surface waves travel in shallow water at a speed of √(gH), where g is the gravitational acceleration (9.8 ms
2) and H is the water depth. For an ocean basin 4 km deep, that’s 720 km h
1; or in a shelf sea with a 40 m water depth, just 72 km h
1, so it takes a tide wave a few hours to cross the Atlantic or to propagate through the North Sea.

1.3 The dance of the Earth and the Moon To begin our journey of understanding fundamentals of tides, consider an Earth with no land and a very deep (deeper than 21 km, in fact), frictionless, ocean that is undisturbed by the atmosphere. Our hypothetical (and quite unrealistic) Earth orbits perfectly in the ecliptic, the equatorial plane of the Sun, and the Moon orbits Earth in a perfect circle over the Equator. There are then only three principal forces acting on the surface of Earth that are of importance to the tide: 1. Earth’s gravitational pull (acting toward the center of Earth); 2. A gravitational force between the Earth and the Moon (on Earth always pointing to the center of the Moon); 3. A “centrifugal force” set up by the rotation of the Earth and Moon around their common center of mass (acting away from the Moon) – but see the next point! Of importance to note here is that the centrifugal force is a fictitious or apparent force, a result of the rotating frame of reference (as opposed to, for an example, a real gravitational force arising from an object with mass). If we ignore Earth’s gravity, this centrifugal force acting away from the Moon would just about balance the gravitational force toward the Moon. We will also neglect the spin of Earth for now, but we will bring it back in later, along with Earth’s gravity.

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Fig. 4.2 Orbits of a simplified Earth-Moon system. This is a schematic of the Earth-Moon motions around their common center of gravity (the black point) if the bodies had equal masses.

The aforementioned centrifugal force arises because the Earth and Moon orbit each other about their common center of gravity, which is located on a line connecting the centers of the two bodies (Fig. 4.2). Because the mass of the Earth is about 81 greater than that of the Moon, in reality the common center of gravity lies within Earth, at a distance of about three-quarters of an Earth radius from the center. Assume – for now and for simplicity – that both bodies have the same mass, so that their center of mass is halfway between them. In a sidereal month, that is, a month measured relative to the fixed stars, the Earth and the Moon orbit this point, but they do not rotate. The motion of the Earth can be visualized if you hold your hand out in front of you and move it around in a small circle with the fingers always pointing in the same direction. Each point on your hand will then move in a circle of the same size. The same happens on Earth, and because all circles traveled are of the same size, each point on Earth experiences the same centrifugal force. Note that the rotation of Earth around its own axis once per day is not related to this and does not contribute to the tide generating force (but it does come in when setting the tidal period). In Fig. 4.2, the paths followed by the small gray and white points during one month are indicated by the dotted circles. Note that both dotted circles are of the same size, and all points on (and within) Earth move in an orbit of the same size each month (remembering all points on your hand, as previously). Consequently, the centrifugal force arising from this motion must be the same at all points on Earth. As we mentioned earlier, the Earth is much larger than the Moon and the common center of gravity lies close to the center of the Earth, but the picture painted previously still applies: each point on and in the Earth moves around the same sized circle in 1 month and experiences an equal centrifugal force.

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Fig. 4.3 The tide generating force. (A) This schematic shows the much simplified (and not to scale) Earth-Moon system. The Earth and the Moon attract each other with the force of gravity (solid black arrows), which is opposed by the fictitious centrifugal force (dashed arrows) acting outward and arising due to the rotation of the Earth-Moon system around its combined center of gravity. (B) The tide generating force arises as the difference between the gravitational pull towards the Moon and the fictitious centrifugal force arising from the joint rotation of the system. The tide generating force at P and Q is indicated by the small thick arrows, which is the difference between the solid black arrows and the dashed arrows. Note that the tide generating force is not equal in direction and magnitude at all points on Earth.

Consider now the force balance on the Moon. The gravitational force there must balance the centrifugal force (Fig. 4.3A); otherwise, the Moon wouldn’t remain orbiting the Earth as it does. We can write this force balance as: GmE mM v2 (4.1) ¼ m M d d2 where G ¼ 6.67  10
11 m3 kg
1 s
2 is the gravitational constant, mE ¼ 5.97  1024 kg is the mass of Earth, mM ¼ 7.35  1022 kg is the mass of the Moon, and d ¼ 384,400,000 m is the Earth-Moon distance. Rearranging Eq. (4.1), the velocity of the Moon in its orbit, v, can be written as: rffiffiffiffiffiffiffiffiffiffi GmE v¼ (4.2) d The period, T, of the Moon’s orbit is now given by the circumference of the orbit (i.e., the distance the Moon will travel, given by 2πd) divided by v, or:

Why is there a tide?

sffiffiffiffiffiffiffiffiffiffi d3 T ¼ 2π GmE

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(4.3)

This is Kepler’s third law and shows that the orbital period of any object orbiting another depends inversely on the mass of the host and on the separation distance to the power of 3/2. On the surface of Earth, the gravitational force on 1 kg of mass is g ¼ GME/R2, where R ¼ 6,371,000 m is the radius of Earth and g ¼ 9.81 m s
2 is the acceleration due to gravity. If we use this in Eq. (4.3), we get: sffiffiffiffiffiffiffiffi d3 T ¼ 2π (4.4) gR2 which is the period of any object in orbit around Earth at distance d from Earth’s center. For the Moon’s orbital period, using the numerical values for R and d shown previously, we get T ¼ 27.4 days. This is in close agreement with the observed orbital period of the Moon around Earth, the sidereal month, which is 27.3 days. However, the Earth also moves 1/12 of the way around the Sun during one sidereal month, and so it takes an additional 1/12 of a month for the Moon to be back in the same place relative to the Sun. This is the synodic month which has a length of 29.5 days, and it is the time between one new Moon and the next.

1.4 The tide generating force As explained previously, the centrifugal force is the same everywhere on Earth, but the gravitational attraction towards the Moon is not (Fig. 4.3A). Newton’s law of gravitaty states that the gravitational attraction between two objects is inversely proportional to the square of the distance between those objects, as in the left-hand side of Eq.(4.1). In accordance with this law, the Moon’s gravitational attraction, F, decreases the farther away from the Moon one gets, so that on the side of Earth facing the Moon (point P, Fig. 4.3B), F is larger than on the far side of Earth, facing away from the Moon (point Q, Fig. 4.3B). Consequently, the centrifugal force and the Moon’s gravity no longer balance everywhere, and this small imbalance is the “tide generating force.” On Earth, on the side facing the Moon (point P), the gravitational pull is greater and the net force is toward the Moon, whereas on the side facing away from the Moon (point Q), the centrifugal force dominates. The effect of this on the ocean is that the ocean is pulled in two directions, towards the Moon in one hemisphere and away from the Moon in the other hemisphere (Fig. 4.3B).

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The forces mentioned previously do balance at one point, however; in the center of the Earth, they are exactly equal. Because the centrifugal force does not change with location yet gravitational attraction does, we can compute the tide generating force as the difference between the Moon’s gravitational pull at the center of the Earth (i.e., a force equal to the centrifugal force but easier to use in the following equations) and the Moon’s gravitational pull on the surface of the Earth. The difference is small – let’s compute the tide generating force, FTG, to see quite how small it is. For simplicity, we consider 1 kg of mass at point P, and with R being the radius of Earth, we get: F TG ðP Þ ¼

GmM GmM 2
d ðd
RÞ2

(4.5)

which is directed toward the center of the Moon. A bit of algebra neatly gets us to: F TG ðP Þ ¼

GmM Rð2d
RÞ d 2 ðd
R Þ2

(4.6)

The ratio of R/d  1/60, so without loss of accuracy, we can take 2d-R  2d and (d-R)2  d2 and get: 2GmM R (4.7) d3 The same calculation can of course be done for all points on Earth, for example, at point Q, the force is directed away from the Moon. We can now use Eq. (4.7) to calculate the magnitude of the tide generating force. To do so, we use the same argument as we did when deriving Eq. (4.4) that GME ¼gR2, and we get:   mM R 3 (4.8) F TG ðP Þ ¼ 2g mE d F TG ðP Þ ¼

As already mentioned, mM/mE  1/81 and R/d  1/60, so FTG/ g  5  10
8 for a 1 kg mass at P. The tide generating force is indeed very, very, small. The reason the force can still generate such massive movements in the ocean is because it is really the horizontal component of the force, the one parallel to the Earth’s surface, which sets up the horizontal motions which drive the tide (consider point S, Fig. 4.3B). The vertical component, perpendicular to the surface is, as we just saw, tiny compared to the Earth’s

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own gravity and only has the effect of making very small local differences to the weight of objects on the Earth’s surface. The residual tidal force acts tangentially to the surface, exerting a tractive force which pulls the sea horizontally rather than lifting it vertically toward the Sun or Moon (a common misconception). The tide generating force varies in strength and direction with changes in both latitude and longitude and is not symmetric about the Equator nor the axis of Earth’s rotation. A detailed derivation of the components of the tide generating force can be found in the literature (Gerkema, 2019; Pugh and Woodworth, 2012), but an example illustration of the geographic variations in the residual tangential components of the force is given in Fig. 4.4. Thus far we’ve been ignoring the Earth’s rotation about its own axis; however, it is the Earth spinning within this pattern of tide generating force which creates tidal motion. As the Earth rotates, and a particular point on the

Fig. 4.4 Tangential components of the tide generating force. Conceptual diagram (not to scale) illustrating the Earth (biggest sphere) and the vectors (gray dashed line) to the Moon (gray sphere) and the Sun (orange sphere) at four different times of year (A–D). In each, the black arrows on the Earth’s surface indicate the relative magnitude and direction in which the residual tidal force is acting. The Sun and Moon are in line; hence, the phase of the Moon in each is New Moon.

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surface moves through this pattern of forces, it experiences a force which can change direction with half-daily and daily rhythms – hence the semidiurnal tides (two high and low waters a day) and diurnal tides (one high and low water a day), respectively.

2. Tidal theories The correlation between the phase of the Moon and the ocean tide was recognized by societies of Ancient Greece (see Chapter 1 for discussion). It was not until the late 17th century, though, that the basis of oceanic tide theory was established by Sir Isaac Newton. Newton explained for the first time that the ocean tide is a result of the gravitational attraction between the Sun and the Moon and the Earth’s water bodies. Known as the equilibrium theory of tides, Newton’s tidal theory described an idealized tide resulting from only lunar and solar forces, ignoring land masses and ocean dynamics. It applied to a hypothetical global ocean in equilibrium with the tide generating force. The equilibrium theory of tides (although limited in scope) is useful because it explains, in simple conceptual terms, the features of the tide most prominent to a casual observer; namely, the fact that there are (i) two tides a day, (ii) a diurnal inequality and (iii) a spring-neap cycle. The equilibrium theory of tides (or sometimes just equilibrium theory), however, fails to account for the timing of the tide and the observed range of the rise and fall of the sea surface. These are important practical details which require a dynamical theory of the tide.

2.1 Equilibrium theory of tides The tide generating force shown in Fig. 4.4 tends to pull (horizontally, remember) the oceans to a point directly beneath the Moon and a second point on the surface of the Earth furthest from the Moon. The result is to create two ’bulges’ in the surface of the ocean, as illustrated in Fig. 4.5. An ocean covering the Earth takes the shape of an ellipsoid, or rugby ball, with the points directed toward or away from the Moon. A point on Earth turning slowly within this ellipsoid ocean will experience one high water when the Moon is directly overhead and another when it is on the opposite side of Earth from that point. And there we have it: two high and low waters per day! We say that the tide is semidiurnal if it has two highs (and two lows) per day… almost. The period of the Lunar tide is actually 12.42 h and not 12 h exactly. This is because in the time it takes for Earth to rotate once, the Moon moves 1/28 of the way around its orbit. That means the Earth will have to rotate an extra 1/28 of a day to catch up with the Moon; so, we must

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Fig. 4.5 Tides and the Moon. Here, we illustrate the shape of the ocean surface predicted by the Equilibrium Theory of Tides. You are looking at the Earth from above the North Pole. The arrows on the Earth represent the tide generating force.

add an extra 50 min, or 0.84 h, to have the Moon directly overhead again. Since the tidal bulges move with the Moon and there are two of them, one will pass by every 12 h 25 min, or every 12.42 h. The equilibrium theory explains the origin of other rhythms of the tide. We showed earlier that the tide generating force is proportional to the mass of the body generating the tide but inversely proportional to the cube of the distance between the objects. The orbit of the Earth around the Sun also creates tidal forces, but these are only about half as strong as those produced by the Moon. The Sun, despite being around 27 million times larger than the Moon (the Sun’s mass, mS ¼ 1.99 1030 kg), only accounts for 46% of the tide generating force because it is much further away from the Earth than the Moon (390 times, in fact, at an average distance of around 150  109 m from Earth). The solar tide has a period of exactly 12 h or half the length of a solar day. Note that the other planets don’t affect the tide for our purposes because of the vast distances to them. The Moon and Sun together create a cycle of larger and smaller tides, which repeats every fortnight (or just over – as explained shortly). At new and full Moons, the Sun and the Moon align (they are in syzygy) and so their equilibrium tide ellipsoids will add up and produce larger than average tides around these times. These larger tides are known as spring tides (note that depending on where you are on Earth, there can be a lag between syzygy and peak spring tides, sometimes referred to as “the age of the tide”). During the Moon’s first and third quarters, the Moon and Sun make an angle of 90 degrees with the Earth (they are in quadrature); their ellipsoids do not align and this results in a smaller than average tide – the neap tide. The spring-neap cycle has a period equal to half of the synodic month, or 14.7 days, because we get spring tides when the Sun and Moon align, and that happens twice every month.

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Fig. 4.6 The origin of the diurnal tidal inequality. When the Moon lies off the plane of the Equator, an observer at O will see two unequal high tides during a period of 24 h.

Another assumption we made at the beginning of this section was that the Moon orbits perfectly over the Equator (so that the tidal bulges are largest there). That is not the case in the real world: the Moon’s orbit is inclined to the Equator at an angle of 28 degrees (see Fig. 4.6) – the lunar declination. This means that the Moon crosses the Equator twice in a month as the declination goes from +28 degrees to
28 degrees and back. Point O on the surface of Earth (Fig. 4.6) will experience two high tides each day because of the Lunar tide, but the one when the Moon is overhead will be larger than the one when the Moon is on the opposite side; there is thus a daily (or diurnal) inequality in the tidal amplitudes. The effect of the diurnal inequality changes during the month as well – it is greatest when the lunar declination is greatest and weakest when the Moon is over the Equator. The same argument can also be applied to the Sun and explains why each year the largest semidiurnal tides appear during the equinoxes, which is when the Sun is over the Equator. So, what do we get if we add these tidal rhythms together? One way of thinking of the diurnal inequality is that it is a semidiurnal tide (with period 12 h 24 min) to which a diurnal tide (with a period of 24 h 48 min) is added. The diurnal tide enhances one of the high waters during the day but diminishes the other, producing the diurnal inequality (see Fig. 4.1). The springneap cycle works the same way, but there we add the lunar and solar semidiurnal tides together (once again, see Fig. 4.1, in particular the tidal signal for Newport, UK). We can think of the resulting tidal signal as a sum of different “components,” which are called tidal constituents. Tidal constituents (conventionally named with a symbol) are mathematically described as pure harmonics, each with its own fixed period. For example, the principal lunar semidiurnal tide is called M2 (M for Moon and the 2 because it is semidiurnal), whereas the Sun’s semidiurnal tide is called S2.

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Table 4.1 Tidal constituents.

Semidiurnal

Diurnal

Long period

Name

Symbol

Rel. amplitude

Period [h]

Principal lunar Principal solar Lunar elliptic Lunisolar Principal lunar Principal lunar Principal solar Lunar elliptic Fortnightly Monthly Semiannual

M2 S2 N2 K2 K1 O1 P1 Q1 Mf Mm Ssa

1.0000 0.4656 0.1915 0.1267 0.5842 0.4148 0.1933 0.0795 0.1722 0.0909 0.0802

12.42 12.00 12.66 11.97 23.93 25.82 24.07 26.87 327.85 661.31 4383.05

The main constituents of the tide and their symbols, relative amplitude, and periods in hours.

The diurnal tide we mentioned previously is described by K1. And then there are about 630 other constituents needed to fully describe the observed tide! Fortunately, we can get pretty far with just the top eight – see Table 4.1 for examples. The concepts discussed previously make up the equilibrium theory. The theory gets the period of the different rhythms of the tide correct, and it provides a decent approximation of the amplitude of the tide at islands in the central Pacific, such as Hawaii (although the theory, in its incompleteness, actually ignores the existence of islands – or any land for that matter). Here, we must also note that it is possible to derive the equilibrium tide from the gravitational potential between the Earth and Moon, and then take into account how the Moon moves around Earth. This is a bit more mathematically challenging and is left for the interested reader – Pugh (1987) covers it in detail. However, the amplitude of the equilibrium tide is about 25 cm, so it obviously doesn’t describe the tide everywhere (see Fig. 4.1). So, what’s missing from this equilibrium theory of the tide?

2.2 Why the tide does not behave as an equilibrium tide There are several reasons why the equilibrium theory fails to describe the real-world tide. Remember that we made some serious assumptions in Section 1: we assumed a deep ocean without continents, and we assumed that the Earth turns slowly enough for the tidal bulges not to be disrupted. However, in order to keep up with the Moon, the tidal bulges have to travel over the surface of the Earth at the same speed as Earth is spinning. The tide is actually a very long period wave (see Section 3) that (for the semidiurnal

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tide) has a wavelength equal to half the Earth’s circumference. As the Earth spins, for the equilibrium tide to hold, this wave has to maintain itself with one crest directly under the Moon. However, there is a problem: because of friction, water waves cannot move fast enough to keep up with Earth’s spin. The tide behaves like a shallow water wave, so its speed is set by √(gH), where H is the water depth (more on this in Section 3.1). The tide is a shallow water wave everywhere on Earth because its wavelength is much greater than the depth of the ocean. The average depth of the ocean is about 4000 m, so the tide travels at an impressive speed of almost 200 m s
1 (on average). But that’s not fast enough: at the Equator, a point on Earth’s surface moves at just over 460 m s
1. This is the first reason why the equilibrium theory “fails”: the tide wave cannot travel fast enough to keep up with the Earth’s rotation and hence cannot maintain equilibrium. The other two reasons the theory does not work in reality are (i) the blocking effect of the continents and (ii) that water moving on a rotating planet is affected by the Coriolis force (see Section 3). As we add increasing complexity (and indeed, reality) to our understanding of the tide, it is time to consider the effects of continents and ocean bathymetry on the tide. Fig. 4.7 shows the global M2 tidal amplitude (which is half the tidal range) simulated by a numerical tidal model, for a series of continental configurations (this is the same model as in Chapters 6–10). In panel A, we can see the equilibrium M2 tide, with an amplitude of less than 30 cm over the Equator. This illustrates that tides on deep water worlds without continents can be represented by the equilibrium theory (Barnes, 2017). The picture immediately changes if we add continents: in panel B, the ocean is still very deep (21 km) and in a “bathtub” configuration (i.e., there is no ocean topography – the sea floor is flat and only rises up to reach the surface at coastlines), and yet we can see amplified tides in Hudson Bay in Canada, in the Red Sea between Africa and the Arabian Peninsula, and in the South China Sea. This is because these basins are now resonant for the M2 tide (see Section 3.3). So, one reason the equilibrium theory of tides fails on Earth is because the continents prevent undisturbed propagation of the tide, but also ocean basins of certain shapes and sizes can amplify the tide (Blackledge et al., 2020). If we take the set-up in panel B but make the ocean 4000 m deep, that is, the average depth of the ocean at present (but it is still a huge bathtub), we get a very large tide everywhere, again because of tidal resonances (panel C). Panel D shows the more realistic, more subdued, but still quite energetic, picture of the present-day ocean tide with ocean bathymetry included.

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Fig. 4.7 From equilibrium theory tides to the real world. Effects of topography on the global M2 tide. The colors show amplitudes (half the tidal range) and saturate at 2 m to show the results more clearly. Land is gray and the black lines show tidal phases separated by 3 h. (A) The equilibrium tide in a very deep ocean (the depth is 21 km everywhere). (B) A 21-km-deep ocean with continents. (C) A 4-km-deep ocean with continents (note that the average depth of the ocean is just under 4 km). (D) The M2 tide for real ocean bathymetry.

2.3 The effects of Earth’s rotation on the tide Here, we introduce the Coriolis force, which can be considered another pseudo force (i.e., a fictitious force, as can the centrifugal force described earlier). The Coriolis force arises from water motion within a rotating (or curvilinear) coordinate system, such as the Earth rotating around its own axis. When viewed from a rotating coordinate system, a large body moving with constant velocity seems to change direction, a result of this apparent Coriolis force (note the motion needs to last for long enough for this to happen, typically a few hours). This pseudo force acts to deflect moving objects to the right (clockwise) in the Northern Hemisphere and to the left (counterclockwise) in the Southern Hemisphere. This effect is called the Coriolis effect. It comes about because at the Equator, a point on the Earth’s surface rotates faster than at the poles. If there isn’t much friction between an object and the Earth, as is the case for the ocean, that object will keep the speed it had at the Equator as it moves northwards, and this will make it appear to be deflected to the right. Similar arguments hold for you moving back toward the Equator after stopping for a bit – you are then moving slower than the Earth beneath you and again, you will be deflected to the right of the

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motion. If you move East or West the same thing will happen – you are deflected to the right (or left in the Southern Hemisphere) because you will feel a different centrifugal force that will shift you to your right. Thanks to the fundamental laws of motion, because we are changing direction in our motion we are accelerating, and that means we are affected by a force – the Coriolis force – which acts on moving objects such as ocean currents and depends on the speed of the current. Tidal currents feel the Coriolis effect and so we must consider this sideways force acting on them. Note that it is NOT the Coriolis force that makes the water spin as you drain the bathtub or sink at home; those motions take place on spatial/temporal scales far too small/short for the Earth’s rotation to have an effect. However, because the tide has a period of 12 h or more, we need to take the Coriolis force into account when describing the tide. The effect turns out to be quite surprising and produces tidal features called amphidromic systems in which the tide wave travels in a sweeping circular motion about a central point (you can see these in Fig. 4.10). There is more on the Coriolis force and amphidromic systems in Section 3; this was a brief introduction for the purpose of explaining the more complete dynamic theory of tides.

2.4 The dynamic theory of tides As has been explained previously, the real tides are far more complicated than those described by the equilibrium theory of tides. Developed to describe the real behavior of the tide, the dynamic theory of tides (or just dynamic theory in the following) was developed by mathematicians and scientists such as Laplace, Euler and Bernoulli during the 18th century and has been further refined with time. This later and more comprehensive theory considers the effects of water depth, bottom friction, inertia, the presence of land, and other factors on the tide. In 1775, Pierre-Simon Laplace introduced his theory of tides, presented to the Royal Academy of Sciences. In 1799, he published this theory (Laplace, 1799), which considered water in motion (as opposed to water in equilibrium – see previously) and expressed the tidal potential as a function of the horizontal forces acting on the tide (which vary temporally and spatially), namely: acceleration ¼ Coriolis force + pressure gradient force + tractive force We can consider the vertical components of the tide generating force to be negligible in relation to gravity (i.e., we consider pressure to be

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hydrostatic, or even more simply put, we assume vertical ocean flow is negligible); the tide is forced by a horizontal tide generating force (the tractive force), the Coriolis force, and the pressure gradient force, which results from slopes in the sea surface. Laplace thus derived what’s known as the Laplace Tidal Equations (Eqs. 4.9), a set of hydrodynamic equations derived from the Navier Stokes shallow water equations (which essentially describe the dynamics of a shallow fluid on a rotating planet). The Laplace Tidal Equations are depth-integrated and apply to the motion of a unit mass of water. They can be written in Cartesian coordinates: ∂u ∂u ∂u ∂η +u + v ¼ fv
g + Fx ∂t ∂x ∂y ∂x ∂v ∂v ∂v ∂η +u + v ¼
f u
g + Fy ∂t ∂x ∂y ∂y   ∂η ∂u ∂v +H + ¼0 ∂t ∂x ∂y

(4.9)

where u and v represent depth-averaged velocities in the x and y directions, respectively (vertical velocities are considered negligible), f is the Coriolis parameter, which varies with latitude, η is the elevation of the sea surface above a reference level, and FX and FY are the tidal forces in the horizontal x and y directions, respectively. Other terms, including friction, can be added as appropriate. The first two equations represent conservation of momentum and the third one represents continuity or conservation of mass. The terms on the left-hand side of the first two equations are accelerations (local and advective). The first term on the right-hand side is the Coriolis acceleration, and the second term is the acceleration produced by the slope on the sea surface, i.e., the pressure gradient force (g is the acceleration due to gravity). These equations form the basis for simulating (modeling) realistic tidal dynamics. One solution to these equations, for an ocean acted on by a periodically varying tidal force, takes the form of a wave, with the tidal period influenced by the rotation of the Earth, and propagating on the ocean surface.

3. Tides in the real world In the present-day open ocean, the maximum tidal elevation amplitude is around 1 m. In accordance with the law of energy conservation, as the tide

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wave approaches the coast and the water depth decreases, tidal amplitudes tend to increase. Thus there can be considerable variations in the nature of shelf sea tides and open ocean tides because of the influence of a shelving seabed, shallow water areas, and the presence and shape of land boundaries. As the tide propagates from the ocean into shelf seas, reflection of the tide wave may occur at the shelf break, within coastal basins, bays, and estuaries. Resonance of the tide wave can occur in basins which have the right depth and width. These resonances result in an amplified tidal signal, producing large tidal ranges such as those seen in the Bay of Fundy, Canada (over 12 m), and in the Bristol Channel, UK (over 10 m).

3.1 The tide as a shallow water wave As explained earlier, the tide is a very long period wave. But what is a wave and why do we refer to the tide as a shallow water wave? Put most simply, a wave is a disturbance in a medium (e.g., air or water) that moves energy without the net transfer of matter. In water, waves propagate through water by causing water particles to oscillate about a fixed position. The wave height is defined as the height of the wave from the trough (bottom) of the wave to the wave crest (top). The wavelength is defined as the horizontal distance between two successive wave crests or troughs. Water waves behave differently depending on the water depth, whether it is considered to be “deep” or “shallow.” The distinction between deep and shallow water waves is determined by the ratio of the wavelength of the wave to the water depth. In terms of wave propagation, waves are considered “deep water waves,” where water is deeper than one half of the wavelength. For a deep water wave, water molecules move in a circular orbit, the diameter of which decreases with distance from the surface and the waves do not interact with the seabed. This circular orbit motion is felt down to a depth of approximately one wavelength; any deeper than this level, the wave’s energy becomes negligible. An example of a deep water wave is a wind-induced ocean wave, generated by either local winds (sea waves) or by distant winds (swell waves). Waves are considered shallow water waves where their wavelength is more than 20 times the water depth (i.e., H < 1/20λ) and their propagation is affected by the sea bed. In these waves, the orbital motion of the water particles is less circular than for deep water waves, becoming increasingly elliptical with depth, as the shallow water waves interact with the seabed. The tide is an example of a “shallow water wave” since, even in the deepest ocean, its wavelength is

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longer than twice the ocean’s depth. But how does this very long period wave propagate around the ocean? As explained in Section 2, the tide generating force changes direction on a 12.42 h cycle (or 12 h for the solar tide) as Earth spins inside the pattern set up by the tide generating force. This means the waves are forced, and there are certain distinct features because of that. Firstly, a forced wave will have the same period as the forcing – 12.42 h for the tide wave in the case of the lunar tide – and will propagate, as we mentioned previously, with a speed, c ¼√(gH). Secondly, the height of the waves depends on the strength of the forcing because the energy in waves increases with the square of their height and here the energy comes from the tidal forcing. The wavelength, L, of a tide wave is the distance between two crests and is given by L ¼ cT, where T is the period and c is the speed given previously. For a water depth of, say, 100 m, the wave speed will be about 30 m s
1, and the semidiurnal tide will have a wavelength of about 1300 km, whereas the diurnal tide will approach 2600 km wavelength. Consequently, tide waves are very long – that’s why they feel the ocean floor as they propagate. Note that, for a wave of given period, both the speed and the wavelength depend only on gravity and water depth.

3.2 Standing and progressive waves In addition to defining waves as either deep or shallow water waves, there are other distinctive properties that can help identify different types of waves. All individual waves can be considered “progressive waves,” where the wave profile travels horizontally, with a wave speed, and inducing a horizontal flow of energy in the direction of wave travel. There are, however, circumstances in which the superposition of two equal progressive waves traveling in opposite directions occurs, creating a “standing wave.” In standing waves, the wave profile remains in a constant position and the vertical oscillations of the water surface occur without horizontal propagation of the wave profile. In the open ocean, the tide tends to propagate as a progressive wave; however, in shelf sea regions, there can be considerable variation in the nature of a tide wave and the tide can be described as either a progressive or a standing wave system (or somewhere in between). Standing waves tend to occur in channels, estuaries, or closed water basins. A seiche in a bay or harbor (or in a bathtub) in which water sloshes back and forth between boundaries is also an example of a standing wave. In reality, few places are purely progressive or standing but can be characterized as mixed- or partially progressive wave systems (Fig. 4.11).

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High water at a point in the ocean or along a coastline occurs when the crest of the tide wave passes that point. Since the speed of the wave depends on water depth, the timing of high water will depend on how fast the wave can travel. For tidal currents in a standing wave system, slack water coincides with low/high water. Conversely, in a progressive (or traveling) wave, that is, a wave where one can see the wave form propagating, flow is in the direction of wave propagation under the crest and in the opposite direction under the trough, and maximum currents occur at high and low water. The maximum current speed, v, beneath the wave is given by shallow water wave theory as: rffiffiffiffiffi g v¼a (4.10) H where a is the tidal amplitude (half the tidal range) and H is again the water depth. In a standing wave (Fig. 4.8B), the elevation is always zero at nodes, and the point of maximum elevation (or amplitude) of the standing wave is called the antinode; these waves are called standing waves because they are horizontally stationary (i.e., they don’t go anywhere), and there is no net transfer of energy. Away from the nodes, the surface goes up and down, but at the nodes, there is no vertical movement. High water occurs to one side of the node when it is low water on the other side and vice versa. In a standing wave, the fastest current speeds occur when the water level is at mid-tide (i.e., halfway between high and low water), and the currents go to zero (slack water) at both high and low tide. A standing wave can be thought of as the sum of two equal progressive waves traveling in the opposite direction. At the nodes, the vertical movement of the two waves always cancel: the crest of one wave is passing at the same time as the trough of the other wave. The currents in the two waves add at the node, because the current are directed with the wave at the crest and against the wave at the trough. At the node, the crest of the one wave meets the trough of the other wave but because the waves are going in opposite directions, their currents are going in the same direction. This means that the maximum currents occur in the locations with no surface signal, and the currents at those points are twice that of those in a single progressive wave (and given in Eq. 4.10). At the coast, where the vertical motion of the sea surface in a standing wave is largest, the currents go to zero at all points in time. These points are known as antinodes.

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Fig. 4.8 Standing vs progressive wave systems. (A) Illustration of how a wave can be reflected at a coast. The tide offshore is then the sum of the incoming and reflected waves traveling in opposite directions (the waves are drawn in side view). Middle and lower panels show the phase relationship between tidal elevations (solid line) and tidal currents (circles) for (B) standing and (C) progressive wave systems.

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3.3 Resonance From Fig. 4.8A, you can see that if an ocean basin is 1/4 of a wavelength long, then you will get a node at the entrance to the basin (the same will be true if the sea is 3/4 of a wavelength long, etc.). In this case, even a very small tide at the entrance to the basin will produce very large tides at the coast: the basin is resonant at the frequency of the tide wave. Resonance can occur within long estuaries too. The same can happen in large basins with a propagating wave as well, but then the size of the basin must be half a wavelength (Arbic et al., 2009; Green, 2010). Resonance occurs in any oscillating system when you force it at its natural period. In the case of the tide, the forcing period is set by the Earth turning within the tide generating force, and the natural period of a basin is set by its size and depth. A quarter wavelength tidal resonance for a tidal constituent with period, T, (here given in seconds) will occur when the basin has a length, x, close or equal to: 1 T pffiffiffiffiffiffi x¼ L¼ gH (4.11) 4 4 Some channels, basins, shelf seas, or even deep oceans may have lengths that match (exactly or some small multiple of ) the period of the tide (or some constituent of that tide). For example, a sea which is 50 m deep would need to be about 240 km long to be in quarter wavelength resonance with the semidiurnal M2 tide. This is the case for the Irish Sea in Fig. 4.9, which explains the large tidal amplitudes in its inner part: the tide is a resonant standing wave in much of the eastern part of the basin (Ward et al., 2018). Resonance may seem like you get something for nothing: a small push produces a big oscillation if it is applied at the right time. We do this when we push a child on a swing: each small push, applied at exactly the right time, provides a small input of energy, which accumulates within the system. The oscillations of the swing increase until the input of energy is matched by energy losses due to friction.

3.4 Coriolis effect, geostrophy, and Kelvin waves The effect of Earth’s rotation on a progressive tide wave is to create a slope of the sea surface at right angles to the direction of propagation, that is, if you look in the direction the wave is moving you will have higher sea level on your right, as can be seen in Fig. 4.10. In panel A, the tidal current is going into the paper through a channel. Consequently, the Coriolis force in the Northern Hemisphere deflects water to the right and piles it up against

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Fig. 4.9 Cotidal charts. (A) Co-tidal chart of the Irish Sea for the M2 tidal constituent. The solid blue lines (co-tidal lines) show times of high water in lunar hours after the Moon’s passage of Greenwich, London, whereas the dashed lines show the tidal amplitude (half the tidal range) m in metres. Also shown are the global phases of the (B) M2 tidal constituent and (C) K1 tidal constituent (data from FES2014 database, https://www. aviso.altimetry.fr).

Fig. 4.10 Rotation and tidal flows. The Coriolis force (CF) produces a slope in the sea surface at right angles to the direction of wave propagation. In the Northern Hemisphere, the slope is up to the right under the crest and up to the left under the trough looking in the direction of wave travel. In each case, the Coriolis force is balanced by a pressure gradient force (PG) produced by the surface slope. Note that the circle with a cross represents a wave going into the paper, whereas the circle with a dot represents a wave coming out of the paper.

the coastline. This in turn sets up a pressure gradient acting from high to low water: gravity is trying to level out the slope by moving water back down the slope. In equilibrium, these two forces will balance each other, so there will not be a net flow across the channel; the cross-channel flow is in geostrophic balance. At low water, in panel B, the current is flowing out of the paper,

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giving a Coriolis force to the right of the flow, leading to a pile up on the left side of the channel (from our point of view) and a steady-state situation in which the water surface slopes down from left to right. The net effect of this is a larger tidal range on the right-hand bank of the channel (looking now in the direction the wave is traveling) and a smaller tidal range on the left-hand bank. In reality, this geostrophic balance occurs within the ocean’s interior, away from bottom and top Ekman layers (a result of wind-driven surface flows), and for horizontal distances greater than a few tens of kilometers and on timescales exceeding a few days. If a progressive tide wave feels the Earth’s rotation and has a coastline to guide it, it is known as a Kelvin wave. Kelvin waves must have a guide – topography or the Equator – on the right-hand side of the direction of wave propagation in the Northern Hemisphere (left in the Southern Hemisphere), and they always have the largest amplitude at the coast. The progressive wave traveling northwards in the Irish Sea in Fig. 4.9A behaves as a Kelvin wave: the tidal range is greater on the Welsh coast than on the Irish coast. When a Kelvin wave is reflected by a coastline, the incoming and reflected waves add to produce an amphidromic system. Instead of the elevations of the two waves – the incoming and outgoing waves – canceling along a nodal line, they do so at a single point: the amphidromic point (see the amphidromic point in the middle of the North Atlantic, Fig. 4.9B). If we look toward the reflecting coastline, the incoming wave will be on the right and dominate on that side, whereas the outgoing wave dominates to the left. Consequently, the high water occurs progressively later moving toward the reflecting coast on the right-hand side of the basin and later moving away from it on the left. The amphidromic points occur at the same places as the nodal lines, that is, 1/4, 3/4, 5/4 of a wavelength from the reflecting coast. If the incoming and reflected waves are exactly the same amplitude, the amphidromic point will lie exactly in the center of the basin. At the amphidromic point, the tidal range is zero, and the range increases as you move toward the coast. Note that although the elevations of the incoming and reflected waves cancel at the amphidromic point, the currents add, so an amphidrome is a place of no vertical tide but strong tidal currents. However, if you look at the amphidromic system in the North sea (or in the North Atlantic in Fig. 4.7D), you can see that the amphidrome does not sit in the middle of the basin. Instead, it is skewed toward continental Europe. This is an effect of friction which we will discuss later.

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3.5 Barotropic and baroclinic tides The tide and other movements of the ocean and atmosphere are sometimes referred to as being either barotropic or baroclinic. Loosely, it is simplest to think of barotropic flows as being two-dimensional, whereas baroclinic flows must be considered three-dimensional. In barotropic conditions, the whole water column oscillates in the same direction at the same time and currents have the same strength and direction throughout the water column. More precisely, in a barotropic fluid, density is a function of pressure only and barotropic flows describe motion in homogeneous (constant density), nondivergent, incompressible fluid. In reality, a true fully barotropic system cannot be achieved – an example of a fully barotropic flow would be an ocean current which is vertically uniform, equal from the ocean surface to the seabed, with constant-pressure surfaces parallel to the sea surface. Examples of (near) barotropic conditions are those found in shallow shelf seas (with full mixing of the water column, e.g., by tidal currents) or in a well-mixed surface layer of the ocean. In contrast, baroclinic flows are depth-dependent, where density varies with depth and with horizontal position. Baroclinic flows occur when surfaces of constant density (isopycnic surfaces) either intersect or are inclined to surfaces of constant pressure (isobaric surfaces). An example of a baroclinic flow is an internal tide resulting from the ocean tide (which is sometimes referred to as the barotropic tide) interacting with steep topography (more on internal tides in Section 4.2). Gravity waves result when a fluid is displaced from a position of equilibrium and can be either surface gravity waves (at the air-sea interface) or internal gravity waves. For surface gravity waves, the ocean’s surface is displaced by the wind and the restoring force is a result of the large density difference between the surface waters and the air. Internal waves are another form of gravity wave which occur within the water body on internal interfaces, for example, when the interface between water masses of different densities is disturbed.

3.6 Tidal currents There are many different types of oceanic currents (where “current” describes the motion of water), such as wind-driven currents, density-driven currents, or tidal currents. An ebbing tide is where the water flows between high- and low water, whereas during a flooding tide, the water flows between low- and high water. Tidal currents are generated by this horizontal flow of water between high and low water (“ebbing tide”), or low and high

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water (“flooding tide”). Tidal currents which ebb and flood in opposite directions are known as “reversing” or “rectilinear” currents. Slack water occurs when these rectilinear tidal currents reverse (switch direction), and there is a period of no (or very low) current velocities, which can last from seconds to several minutes. More frequently though, especially offshore, a vector representing the speed and direction of the current will turn with the tide such that the tip of the vector traces out an ellipse. As with the tidal elevations, tidal currents are periodic by nature, flooding and ebbing with the rise and fall of the tide, although slack water does not always coincide with low or high tides (see Section 3.2). Tidal currents also display periodicity with the phases of the Moon. For spring tides, when the vertical variation in the tide is greatest, more water must flow between high and low waters (and vice versa) and so relatively fast tidal currents (“spring currents”) occur. During neap tides, “neap currents” occur, which are slower than spring currents, since less water must flow between the relatively lower tidal amplitudes.

3.7 Tidal charts Tidal charts are maps showing the spatial properties of the tide in a specific ocean basin. Locations where high water occurs at the same time are joined by lines called cotidal lines (which link locations which have the same tidal phase). In a basin with a progressive wave, the cotidal lines look like a series of roughly parallel lines showing the timings of high tide as the wave propagates. Fig. 4.9 shows cotidal charts for the M2 constituent for the Irish Sea (panel A), and the M2 (panel B) and K1 (panel C) phases in the global ocean. In panel A, we can see approximately parallel lines between Ireland and Wales, with the relative timing of high tide, shown by the numbers on the line, appearing later and later as we move north. The numbers show time of high water relative to the Moon’s passage over the Greenwich meridian expressed as lunar hours; one lunar hour is roughly 1 h 2 min. This shows that the tide in this part of the Irish Sea is a progressive wave coming from the south and moving north. We can see from the figure that high tide at the entrance to the Irish Sea occurs 6 h behind the Moon’s passage over the Greenwich meridian, so if the Moon passes this meridian at 12 noon, high water will be at 1800 in the southern Irish Sea. Why are there no cotidal lines in the Northern part of the Irish Sea? This suggests that high tide takes place everywhere at the same tide (or that there is no tide; for the Irish Sea, this is not the case), at about 10.5 h after the

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Moons passage over Greenwich; this is the pattern we get from a standing wave. The incoming wave is reflected at the English coast and creates a standing wave pattern (see Fig. 4.11). The tidal range is shown by the dashed lines in Fig. 4.9A. These lines are called co-range lines and show how the tidal range varies between points. Note that, in the Irish Sea, the tidal range increases steadily from the south toward the north and is greatest against the English coast because of the standing wave system in the region. The smallest tidal range in the Irish Sea is found around south-west Ireland, where it looks like the cotidal lines converge at a point inland (a “degenerate” amphidromic point, a result of friction – more on this in Section 4.1). In Fig. 4.9B, we see an obvious point in the North Atlantic where the phase lines converge (or we could see it as the phase lines radiating out from that point) – again, this is an amphidromic point, as mentioned previously. At these points, there is no tidal range so the phase is always the same.

4. Tidal energetics and energy losses It takes work to keep the Earth spinning within the pattern generated by the tide generating force. The tide acts as a brake, slowing the Earth’s spin; eventually (after a very long time), the tide will bring the Earth to rest with the same hemisphere always facing the Moon. This has already happened to the Moon itself: the Earth’s tidal forces have long ago “tidally locked” the Moon’s spin so that we always see the same face of the Moon. It is estimated that the rate at which energy is lost from the Earth’s spin is around 3.7 TW, 3.5 TW of which is accounted for from ocean tides (the remainder is lost to friction from tides in the atmosphere and solid Earth; see Chapters 15 and 16 for discussions) (Egbert and Ray, 2001).

4.1 Tidal friction Because of tidal friction, the Earth’s spin is gradually slowing down and the day is getting longer at a rate of 2 ms
1 per century at present (Bills and Ray, 1999). As the Earth slows, its angular momentum is passed to the Moon, which as a result, is moving further away from Earth at a measurable rate of about 3.8 cm per year (Daher et al., 2021; Williams and Boggs, 2016). We will come back to these effects in later chapters, when we describe how the tide has changed throughout Earth’s history. The frictional force between a current and the sea bed is observed to increase with the square of the flow speed – the friction is said to be quadratic. The rate at which energy is lost is proportional to the force multiplied

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by the flow speed and so depends on the cube of the speed. A current of 1 m s
1 will lose one million times more energy through friction than a current of 1 cm s
1. For this reason, most tidal energy losses occur in the resonant shelf seas, where the currents are fast. Tidal energy is created in the deep ocean, where the tidal forces act on the great mass of water, but it is lost (mostly, but not completely as we shall see) around the edges of the ocean through friction by fast currents flowing in shallow water. At a local level, tidal friction has noticeable effects on the way that the tide behaves. Because the energy in a wave depends on its amplitude, the amplitude in a progressive wave decreases the further the tide is propagating – it becomes damped. If the tide is reflected off a coast, a standing wave will form near the coast where the incoming and reflected waves have roughly the same amplitude. Further from the coast, however, frictional damping makes the incoming wave bigger than the reflected wave and the sum of the two waves looks more like a progressive wave traveling toward the coast. This is what happens in the Irish Sea: in the northern part of the Irish Sea, close to the reflecting English coast, the reflected wave is energetic enough to produce a standing wave. However, farther away from the English coast, in the channel between Wales and Ireland, the reflected wave is weak, and the tide behaves as a progressive wave traveling northwards up the channel, before entering into the standing wave system in the northern Irish Sea (Fig. 4.11). In a Kelvin wave (Section 3.4) with no friction, the incoming and reflected waves cancel each other exactly in the middle of a basin. In the real world, however, friction reduces the amplitude of the reflected wave compared to the incoming wave. They no longer cancel in the middle of the basin; there is a shift in the position of the amphidromic point toward the left-hand coast (the one the outgoing wave is propagating along) as you look into the basin (see Fig. 4.12 for a schematic) (Brown, 1987; Taylor, 1922). Note (Fig. 4.9) that there is a suggested amphidromic point just inside the eastern coastline of Ireland; this is known as a virtual or degenerate amphidrome. These come about because the frictional effects have been so strong that the amphidromes have shifted out of the sea altogether and on to land.

4.2 Internal tides Although tides in the ocean are generally much smaller than those in shallow seas, the surface tides lose energy through the creation of waves on layers of different densities in the interior of the deep ocean. Where these internal

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Fig. 4.11 Tidal characteristics in a semienclosed shelf sea. Time difference (in hours) between high water and peak tidal current speeds of the M2 tidal constituent in the Irish Sea. The time difference is given by the colorscale, with color contours for every 10 min. In the standing wave system, peak current speeds occur around mid-tide (maximum time difference shown here), whereas in the progressive wave system, peak tidal current speeds coincide with high water.

waves have the same period as the tide, they are called internal tides, or the baroclinic tide (see e.g., Garrett, 2003 for an overview). These internal tides are an important feature of the tidal system and are generated by the interaction of the tidal flow with (steep or rough) topography such as sea mounts and (strong) stratification. Energy is transferred from the surface tide to the internal tide, resulting in energy losses from the barotropic to the baroclinic tide (see, e.g., Nycander, 2005; Vic et al., 2018 for the theoretical

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Fig. 4.12 Friction and amphidromes. The schematic shows how friction moves the amphidromic point toward the left-hand shore, looking into the basin. (A) shows how the amplitude of the incoming and outgoing wave vary across the basin at the node. Here, there is no friction so the amplitude of the two waves is equal, meaning that they cancel in the center of the basin. (B) If friction is acting on the flow, it will reduce the amplitude of the outgoing wave and the position of the amphidrome is displaced to the left, that is, toward the coast guiding the outgoing wave.

discussions). While some energy is dissipated locally through bed friction at the internal tide generation site, these internal waves can propagate for thousands of kilometers from their generation site until they eventually break and dissipate their energy remotely (Zhao et al., 2016; Alford, 2003). Internal tides play an important role in maintaining the deep circulation of the ocean, a key part of the climate control system of our planet (Munk, 1966; Wunsch and Ferrari, 2004). The ocean is warmed by the Sun at the Equator and water flows in surface currents toward high latitudes, losing heat to the atmosphere as it does so. In polar regions, water cools, sinks to the sea floor, and returns to equatorial latitudes as a deep current. On its way, the deep water is mixed back up to the surface throughout the ocean to complete the circulation. It takes energy to perform this mixing and a major part of the energy is supplied by the internal tides. The amount of mixing needed to maintain the deep ocean circulation is small (it has been compared to a standard kitchen food mixer whirling away in each cubic kilometer of ocean), but it is required, nevertheless. It is difficult for turbulence generated at the boundaries of the ocean (the surface and the sea floor) to reach the interior of the open ocean, but the turbulence generated by the vertical shear in the internal tide is created within the body of the ocean and is a key mixing agent; the consequences of this mixing will be discussed in later chapters.

5. Chapter summary 1. The ocean tide is created by the tide generating force, the difference between the Moon’s gravitational pull at a point on the surface of the

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Earth and the centrifugal force needed to maintain that point in orbit about the Moon. The tide generating force points toward the Moon on the side of the Earth directly below the Moon and away from the Moon on the opposite side. A similar (and smaller) tide generating force is created by the Earth’s orbit about the Sun. If the Earth were covered by a deep ocean, water would flow under the influence of the tide generating force. Two bulges in the ocean would be created, one directly below the Moon, the other on the opposite side of the Earth to the Moon. As the Earth turns once a day within these bulges, a point on the Earth’s surface would experience two high tides each day. This is Newton’s equilibrium theory of tides. It accounts for many of the observed features of the tide, but not for two important practical aspects: the height of the tide at a particular location and the timing of high waters. The equilibrium theory of tides fails because the ocean is not deep enough for a wave on the surface to maintain its crest beneath the Moon as the Earth spins. In addition, continents block the progression of the tide wave, which is also affected by the spin of the Earth and friction. These processes can be included in a dynamic theory of tides. The solution to the dynamic equations for an ocean forced by an oscillating tide generating force is a series of waves. The waves (which have the same period as the forcing) have long wavelengths and behave as shallow water waves, that is, their speed just depends on the water depth. When a tide wave travels from the ocean onto the continental shelf, the amplitude increases as the energy is compressed into the much shallower water. In this form, the tide wave travels as a progresisive wave. The wave is also reflected at the coast and the combination of the incoming and reflected waves can create a standing wave. The amplitude of the standing wave decreases from a maximum at the coast to zero at a nodal line where the vertical movement in the two waves traveling in opposite directions, cancels. If the shelf has the right dimensions such that the nodal line lies near the shelf break, there can be a great amplification of the tidal range from the ocean to the coast. The shelf is said to be in tidal resonance. Tide waves are affected by Earth’s rotation. For a wave traveling up a channel in the Northern Hemisphere, the amplitude increases towards the right when looking in the direction of wave travel. Such a wave is termed a Kelvin wave. When a Kelvin wave reflects at a coast, it creates a pattern referred to as an amphidromic system. The crest of the wave sweeps around a central point of no tide (an amphidromic point). The tidal

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range increases outwards from this point. Amphidromic systems are found in all of the oceans and in many shelf seas. 6. Friction between tidal currents and the sea bed removes energy from the tide. At a local level, this can affect the tidal behavior: amphidromic points are moved by friction. Globally, tidal friction is removing energy from the Earth’s spin: the day length is slowly increasing. Most tidal friction occurs in shelf seas, where currents are fastest but a significant drain of tidal energy occurs in the deep ocean where the surface (or barotropic) tide passes energy to the internal (baroclinic) tide.

References Alford, M.H., 2003. Redistribution of energy available for ocean mixing by long-range propagation of internal waves. Nature 428, 159–162. https://doi.org/10.1038/ nature01591.1. W0018 2003;21:159–63. Arbic, B., Karsten, R., Garrett, C., 2009. On tidal resonance in the global ocean and the back-effect of coastal tides upon open-ocean tides. Atmosphere - Ocean 47 (4), 239–266. https://doi.org/10.3137/OC311.2009. United States. Arnold, A.K., Lewis, H.W., Hyder, P., Siddorn, J., O’Dea, E., 2021. The sensitivity of British weather to ocean tides. Geophys. Res. Lett. 48 (3), e2020GL090732. Barnes, R., 2017. Tidal locking of habitable exoplanets. Celest. Mech. Dyn. Astron. 129 (4), 509–536. https://doi.org/10.1007/s10569-017-9783-7. United States: Springer Netherlands. Bills, B.G., Ray, R.D., 1999. Lunar orbital evolution: a synthesis of recent results. Geophys. Res. Lett. 26 (19), 3045–3048. https://doi.org/10.1029/1999GL008348. United States: American Geophysical Union. Blackledge, B.W., Green, J.A.M., Barnes, R., Way, M.J., 2020. Tides on other earths: implications for exoplanet and palaeo-tidal simulations. Geophys. Res. Lett. 47 (12). Brown, T., 1987. Kelvin wave reflection at an oscillating boundary with applications to the North Sea. Cont. Shelf Res. 7 (4), 351–365. https://doi.org/10.1016/0278-4343(87) 90105-1. undefined. Daher, H., Arbic, B.K., Williams, J.G., Ansong, J.K., Boggs, D.H., M€ uller, M., Schindelegger, M., Austermann, J., Cornuelle, B.D., Crawford, E.B., Fringer, O.B., 2021. Long-term Earth-Moon evolution with high-level orbit and ocean tide models. J. Geophys. Res. Planets 126 (12), e2021JE006875. Egbert, G.D., Ray, R.D., 2001. Estimates of M2 tidal energy dissipation from TOPEX/ Poseidon altimeter data. J. Geophys. Res. Oceans 106 (10), 22475–22502. https:// doi.org/10.1029/2000jc000699. United States: Blackwell Publishing Ltd. Garrett, C., 2003. Internal tides and ocean mixing. Science (80-) 301, 1858–1859. Gerkema, T., 2019. An Introduction to Tides. Cambridge University Press. Green, J.A.M., 2010. Ocean tides and resonance. Ocean Dyn. 60 (5), 1243–1253. https:// doi.org/10.1007/s10236-010-0331-1. United Kingdom. Horsburgh, K.J., Wilson, C., 2007. Tide-surge interaction and its role in the distribution of surge residuals in the North Sea. J. Geophys. Res. Oceans 112 (8). https://doi.org/ 10.1029/2006JC004033. United Kingdom: Blackwell Publishing Ltd. Laplace, P.S., 1799. Traite de Mecanique Celeste, II. Duprat. Paris. Munk, W.H., 1966. Abyssal recipes. Deep-Sea Res. Oceanogr. Abstr. 13 (4), 707–730. https://doi.org/10.1016/0011-7471(66)90602-4. United States.

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Nycander, J., 2005. Generation of internal waves in the deep ocean by tides. J. Geophys. Res. 110, C10028. https://doi.org/10.1029/2004JC002487. Pugh, D., 1987. Tides, Surges and Mean Sea Level: A Handbook for Engineers and Scientists. John Wiley & Sons, Chichester, p. 472. Pugh, D., Woodworth, P., 2012. Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes, Sea-Level Science: Understanding Tides, Surges, Tsunamis and Mean Sea-Level Changes. Cambridge University Press, United Kingdom, pp. 1–395, https://doi.org/10.1017/CBO9781139235778. Rulent, J., Bricheno, L.M., Green, J.A., Haigh, I.D., Lewis, H., 2021. Distribution of coastal high water level during extreme events around the UK and Irish coasts. Nat. Hazards Earth Syst. Sci. 21 (11), 3339–3351. Simpson, J.H., Bowers, D., 1981. Models of stratification and frontal movement in shelf seas. Deep Sea Res. A, Oceanogr. Res. Pap. 28 (7), 727–738. https://doi.org/10.1016/01980149(81)90132-1. United Kingdom. Simpson, J.H., Hunter, J.R., 1974. Fronts in the Irish Sea. Nature 250 (5465), 404–406. https://doi.org/10.1038/250404a0. United Kingdom. Taylor, G.I., 1922. Tidal oscillations in gulfs and rectangular basins. Proc. Lond. Math. Soc. s2-20 (1), 148–181. https://doi.org/10.1112/plms/s2-20.1.148. Wiley. Tsunogai, S., Watanabe, S., Sato, T., 2016. Is there a ‘continental shelf pump’ for the absorption of atmospheric CO2? Tellus Ser. B Chem. Phys. Meteorol. 51 (3), 701–712. https:// doi.org/10.3402/tellusb.v51i3.16468. Informa UK Limited. Vic, C., Garabato, A.C.N., Green, J.A.M., Spingys, C., Forryan, A., Zhao, Z., et al., 2018. The lifecycle of semidiurnal internal tides over the northern mid-Atlantic ridge. J. Phys. Oceanogr. 48. https://doi.org/10.1175/JPO-D-17-0121.1. Ward, S.L., Robins, P.E., Lewis, M.J., Iglesias, G., Hashemi, M.R., Neill, S.P., 2018. Tidal stream resource characterisation in progressive versus standing wave systems. Appl. Energy 220, 274–285. Williams, J.G., Boggs, D.H., 2016. Secular tidal changes in lunar orbit and earth rotation. Celest. Mech. Dyn. Astron. 126 (1–3), 89–129. https://doi.org/10.1007/s10569016-9702-3. United States: Springer Netherlands. Wunsch, C., Ferrari, R., 2004. Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36, 281–314. https://doi.org/10.1146/annurev. fluid.36.050802.122121. United States: Annual Reviews Inc. Zhao, Z., Alford, M.H., Girton, J.B., Rainville, L., Simmons, H.L., 2016. Global observations of Open-Ocean Mode-1 M 2 internal tides. J. Phys. Oceanogr. 46, 1657–1684. https://doi.org/10.1175/JPO-D-15-0105.1.

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SECTION 2

A tidal journey through time

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CHAPTER 5

A timeline of Earth’s history João C. Duarte

Geology Department and Instituto Dom Luiz (IDL), Faculty of Sciences of the University of Lisbon, Lisbon, Portugal

1. Geological time The arrow of time is one of the most fascinating, yet inexplicable, characteristics of our Universe. Contrary to the more intuitive space dimensions, time runs only in one direction: from the past into the future. We, humans, live in that tiny glimpse of which we call present. To understand the world around us, we need to recall past events and keep a memory of them. We need this memory to predict and prepare for the future. All our activities, including science, depend on this aptitude. But while we can easily keep the memory of episodes that we have physically experienced, how can we learn about events that happened too far deep in time, in places where no one has ever been before? For that, we must rely on a record – the geological record. The geological record comprises the totality of all rocks that have formed over geological time and that have been preserved. This includes, for example, sedimentary basins and volcanic rocks, as well as fossils of past life and organisms. The geological record is always incomplete, but it does not come about randomly. The layers of rocks are organized in an orderly way, and their disposition obeys basic principles. Old rocks are generally buried below younger ones, and disruption reveals that a certain event occurred. Like detectives at a crime scene, we use these clues to reconstruct past events and learn about the processes that cause them. The geological time is the interval of time occupied by the geological history of the Earth (Encyclopaedia Britannica, 2022). Technically, it starts when the planet formed 4600 million years ago and extends all the way into the present. Given the immensity of time involved and the fact that geological processes are slow, scientists consider the basic unit of time to be the million years. Very much like one uses hours and days to organize events in daily life, Earth scientists typically deal with millions or tens of millions of years to describe major changes that occurred deep into the Earth’s

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geological time. It is not that there are no processes acting at shorter time scales, but these are rare and may not leave a direct and/or a widespread record. Instead, they are usually individual events that are difficult to correlate, such as a local landslide or the erosion of a beach shore. Tidalites may be another example (see Chapter 12). However, sometimes these short timescale events have global impacts and leave a unique mark in the geological record. An example is the impact of a meteorite, the eruption of a supervolcano, or the simple disappearance of a widespread fossil. This leads to an important concept for those working with geological time: correlation. To build useful time scales, we must be able to correlate unique events that have happened simultaneously in different locations on Earth. But how do we do this? Was it always that simple to measure geological time? How do we know the age of the Earth? The Greeks thought the Earth was eternal and that it had no beginning. Subsequent theological theories of creation implied that the Earth was formed sometime in the recent past. James Ussher (1581–1656), based on accounts from the Bible, calculated that the Earth was created on October 23, 4004 BC, approximately 6000 years ago. In the 18th century, several thinkers started to speculate that the Earth could be older. Among them were Benoıˆt de Mailler and Gorge-Louis Leclerc (also known as Buffon), who conjectured that the age of the Earth could be of the order of a few million years. Around the same period, James Hutton, one of the founders of geology, had a clear vision of the immensity of time. He understood the meaning of geological unconformities and that geological processes were extremely slow and cyclical. In his landmark book Principles of Geology, he opened the way to the use of geological strata as the basis for a geological time scale. It is worth noting that this was the ground that allowed Charles Darwin to develop his ideas of biological evolution and natural selection, processes that required an enormity of time. According to Hutton, however, there was no clear evidence of a beginning of time, which lead him to defend a uniformitarian perspective. Nevertheless, using estimates of erosion and sedimentation, he concluded that the geological record went back at least a few hundreds of millions of years (Galopim de Carvalho, 2014; Levin and King, 2016). Several authors have since tried to calculate the age of the Earth using different methods. John Phillips (1800–1974) used sedimentation rates and arrived at a value of 96 million years. James Croll (1821–90) used a combination of strata and fossils from glacial periods and calculation of orbital parameters and arrived at a value of 240 million years. Samuel Haughton

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(1821–97), based on calculations of heat loss and climate change, arrived at values of the order of 2300 million years for the formation of the oceans. However, these calculations were highly uncertain. Just as an example, William Thomson (1824–1907), also known as Lord Kelvin, using calculations of the cooling of the Earth arrived at values of the order 20–40 million years. Also worth mentioning is the calculation of the age of the oceans by John Joly using the concentration of salt, arriving at an age of the order of 80–100 million years (Galopim de Carvalho, 2014; Levin and King, 2016). However, it was only after the discovery of radioactivity that significant progress was made, for two reasons. Radioactivity revealed that the cooling of the Earth was not as simple as previously thought. The Earth is still heating due to radiogenic heating of the mantle. Second, radioactivity provided an effective tool for measuring time with unprecedented precision. But first, let’s go back again and understand how geologists organized the geological record, and with it, time itself.

2. Chrono-stratigraphy In the 17th century, Niels Steensen (1638–96), also known as Nicolas Steno, laid the grounds for what become known as stratigraphy. Steno presented simple principles that were key to the organization of the geological record by observing sedimentary strata. The first is the principle of original horizontality, which states that all the sedimentary strata are deposited horizontally. If a layer is not horizontal, it is because a later process acted on it (e.g., an event that led to folding). The second principle is that of original lateral continuity, which states that a layer extends in all directions until it thins out at the limits of the basin. If a layer is abruptly interrupted, it is because a subsequent event/process acted on it, such as the formation of a fault. The third, and most important, is the principle of superposition, which states that in an originally sedimentary sequence the oldest layer is at the bottom and that the layers become increasingly younger toward the top of the sequence. This became known as the Fundamental Principle of Stratigraphy (Levin and King, 2016). These all seem simple and obvious principles, but at the time, they were revolutionary. After all, the biblical flood would have left a disorganized geological record. These orderly successions of sedimentary layers had major implications: chrono-stratigraphy was born. A few years later and based on geological observations in northern Italy (Fig. 5.1), Giovanni Arduino (1714–95) proposed the division of the Earth’s history into four periods: Primitive, Secondary, Tertiary, and Volcanic, later

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Fig. 5.1 Giovanni Arduino’s stratigraphic section in the province of Vicenza (1758). (Public Domain.)

named the Quaternary. The primitive was evidenced by hard rocks that occurred at the core of mountains, such as granites, schists, and gneisses. The Secondary left a record of stratified limestone rocks with marine fossils. Tertiary layers were made of less consolidated rocks, sometimes with marine shells. And finally, the Volcanic period corresponded to the rocks formed by actual vulcanism, as well as river sediments and beach sands (Galopim de Carvalho, 2014; Levin and King, 2016). It is interesting that some of this terminology persists today, in particular, the Tertiary and the Quaternary. It was William Smith (1769–1839) who understood that different strata had different fossil assemblages and that this holds true even when the type of rock is different. He did not know why this was the case – Darwin still had to present the theory of evolution – but it was the key to correlate different strata worldwide. This became known as the principle of fossil succession and was the basis for what become known as biostratigraphy, which allowed the development of relative geochronology. The last key of the puzzle was introduced by Charles Lyell (1797–1875), as the principle of cross-cutting relationships, which states that if a rock is faulted or intruded by another rock, the fault activity, and the intrusive rock postdate the original rock (Galopim de Carvalho, 2014; Levin and King, 2016). This was of great importance to organize the geological record in complex deformed regions,

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where the fossil record is often missing. An extension of this principle is the principle of inclusion. If a rock contains another rock, the containing rock is older than the rock contained. A few years later, inspired by Lyell’s book Principles of Geology, Charles Darwin (1809–82) developed a theory that allowed explaining why rocks of different ages have different fossil assemblies – the theory of evolution. It wasn’t until the discovery of radioactivity by Joseph Barrel (1871–1937) that absolute geochronology was possible. This was based on the evidence that some radioactive isotopes decay at a precise rate over millions of years. And, therefore, knowing the initial amount of the parent isotope and the amount of the daughter elements at the present, geologists can calculate with great precision the age of the respective rock (Levin and King, 2016). Finally, a powerful tool was available to quantify geological time. But can we tell the age of the Earth? Absolute geochronology, using isotope geology, allowed dating the oldest pieces of geological record ever discovered. These are just a few grains of zircon preserved inside younger rocks. Zircons are known for being almost indestructible. The oldest of these grains were found in Western Australia and yielded an age of 4400 million years (Wilde et al., 2001). However, geologists think that the Earth must be older and estimate its age using the dating of primitive meteorites that are thought to have formed around the same time as the Earth. These yield an age of 4560 million years (Baker et al., 2005).

3. The geological timescale The development of geological time scale was a slow process that started much before absolute dating was possible. For this reason, much of the nomenclature we use is still inconsistent, sometime based on the name of localities and regions, other times based on the type of rocks or named after an event (Levin and King, 2016). Notwithstanding, in the last years, geologists have made an effort to put together a consistent timescale (Fig. 5.2). The geological timescale is organized in Eons, Eras, Periods, and Epochs, very much like a human calendar is organized in years, months, weeks, and days. The main divisions usually correspond to major changes in life forms, often extinctions, which most of the time correlate with major global processes such as climate changes, meteoritic impacts, volcanic eruptions, glaciations, and formation or break-up of supercontinents. There are four Eons: the Hadean, the Archean, the Proterozoic, and the Phanerozoic. In the next

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Fig. 5.2 Table of geological time. Adapted from the International Commission on Stratigraphy (ICS) (Cohen et al., 2013).

section, we will briefly detail the main events that occurred in each of these Eons. For this, we will refer to the most recent table produced by the International Commission on Stratigraphy (ICS; Cohen et al., 2013).

4. Main events in Earth’s history 4.1 The Hadean Eon (4600–4000 Ma) The Hadean is an informal division of the Earth’s history of which there is no significant rock record. Its beginning corresponds to the formation of the Earth around 4600 million years ago and ends with the start of the Archean Eon 4000 Ma. The word Hadean derives from the Greek god Hades, the king of the underworld.

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Earth is thought to have formed from the accretion of space debris and heavy meteorite bombardment (Yu and Jacobsen, 2011; Chambers, 2004). Their collisions produced kinetic energy, which, together with radiogenic heating, led to the partial melt of the embryonic planet. These led to the first episode of differentiation of the Earth and the formation of a metal core (Li and Agee, 1996). During this period, the Earth likely did not have a stable lithosphere or a differentiated crust. The initial materials that composed the Earth are thought to have contained a significant amount of water (Drake, 2005) and soon after it formed a primitive atmosphere developed as the result of volatiles out-gassing, mostly CO2, hydrogen, and water vapor. The water vapor condensed in an ocean, leaving behind an atmosphere mostly made of CO2. The elevated atmospheric pressure allowed the ocean to persist with temperatures at the surface well above 100 °C. Zircons suggest that a liquid water ocean already existed 4400 million years ago (Wilde et al., 2001), and the ocean volume probably did not change much since. This means that during this period, Earth may have transitioned from a global magma ocean to a relatively solid lid. The Moon is thought to have formed around 4500 million years, by one giant impact with a Mars-sized proto-planet named Theia (Canup and Asphaug, 2001; Halliday, 2008). Immediately after its formation, the Moon was located around 100,000 km away from the Earth, just outside the Roche limit, the minimum orbit below which it would have been pulled apart toward the Earth. The impact of Theia and Earth melted both bodies and led to their partial fusion. Their proximity caused very strong tidal forces that caused a rapid recession of the Lunar orbit. By the time oceans formed (or re-formed) around 4100 Ma, the Moon was already 275,000 Km away from the Earth. This relative proximity generated a strong equilibrium tidal force.

4.2 Archean Eon (4000–2500 Ma) The Archean starts 4000 Ma and extends all the way until 2500 Ma. During this eon, fundamental changes occur in our planet. These include the formation and growth of continents, the emergence of some kind of tectonic regime, and the appearance of life itself. The term Archean has the meaning of “beginning” or “origin,” as the Archean used to be the first of the Eons, before the Hadean was recently introduced. It is likely that, in the Hadean and in Archean, Earth did not exhibit a modern style of plate tectonics, although there is still strong debate about

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what the geodynamic regime of the early Earth looked like. Most authors believe that some sort of sluggish tectonics without a global interconnected plate mosaic and proper subduction zones existed by that time (e.g., Lourenc¸o et al., 2018; Stern, 2018). Others, on the other hand, argue that plate tectonics initiated very quickly and has been recycling material on Earth for billions of years (see Korenaga, 2013). Notwithstanding, recent studies suggest that an important shift occurred 3800–3600 Ma, leading to the generation of some kind of proto-subduction zones and recycling of the lid (Bauer et al., 2020) and that 2500 Ma a modern style of plate tectonics was already operating (Palin and Santosh, 2021). However, this is still disputed, and other authors suggest that modern plate tectonics was only effective in the last 1000 Ma (see Korenaga, 2013; Stern, 2018). The debate is complex and ongoing. In part, it is related to how we define plate tectonics and if we use a more or less restrictive definition. There are also proposals that plate tectonics had an early start and then went through a period of quiescence (the Boring Billion), after which restarted again (Silver and Behn, 2008). There is also a debate on the extent of the Archean continental crust, although it seems certain that a few continental fragments started to emerge during this period (see e.g., Stern, 2018). Some authors even argue that the first supercontinents might have formed during the Archean: Vaalbara at 3700 Ma (de Kock et al., 2009; Mahapatro et al., 2011), Ur at 2600 Ma (Rogers, 1996), and Kernoland at 2300 Ma (Bradley, 2011; Nance et al., 2014). The land to ocean ratio on Archean Earth was likely much lower than today, with land making up less than 15% of the planet’s surface compared to 30% today. These big oceans behaved much like the Pacific Ocean, being far too large to host any kind of tidal resonance. However, the Archean tide is still quite energetic simply because of the proximity of the young Moon (see Chapter 6). We can only speculate as to what the Archean (and Hadean) climate was like, but it was likely unrecognizable. The surface temperature was hotter, around 40 °C, and there was very little free oxygen (Catling and Zahnle, 2020). This means there was no ozone layer so Earth’s surface would have been baked with UV radiation. This may have been countered by high levels of sulfur in the atmosphere absorbing the UV radiation, but the surface would have still been largely inhospitable. Life existed in the Archean oceans; however, it is uncertain when it exactly started. There is some evidence that it might have emerged in the Hadean, 4100 Ma, based on potential biogenic carbon found in zircons

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(Bell et al., 2015). However, it is only certain that life existed 3700 Ma, based on biogenic graphite found in metamorphosed sediments in Western Greenland (Ohtomo et al., 2013). The first direct fossils are stromatolites, which are microbial mats made of bacteria and cemented rocks, found in Western Australia and dated from 3480 Ma (Noffke et al., 2013). The development of photosynthesis likely occurred at around 3500 Ma (Anbar et al., 2007), and the first terrestrial organisms may have developed as far back as 3200 Ma (Homann et al., 2018; Beraldi-Campesi, 2013). It is also not known where life started. Some theories suggest that it might have been near hydrothermal vents in the Archean oceans, fed by chemicals and heat from the deep Earth, or in microscopic rock cracks formed by volcanism or impact events, protected from the harsh Archean environment (Martin et al., 2008). Other hypotheses propose that life may have developed on the Archean shores, where the tide going in and out allowed chemicals in the seawater to be mixed by wave action (repetitively concentrated and diluted) encouraging chemical reactions necessary for the formation of life (Lathe, 2004). Notwithstanding, life during the Archean was limited to single-celled organisms (prokaryotes) and included mostly bacteria and other organisms of the domain Archaea. Cyanobacteria were essential for the creation of an oxygenized atmosphere in the next Eon, the Proterozoic (Stanley, 1999).

4.3 Proterozoic Eon (2500 Ma–541 Ma) The Proterozoic is the longest of the Eons, spanning from 2500 to 541 Ma, and corresponds to the emergence of the modern world. It was during this period that the atmosphere become oxygenated, the current plate tectonics regime was established, and multicellular life emerged. The Proterozoic is generally divided into three eras: Paleoproterozoic, Mesoproterozoic, and Neoproterozoic. The term Proterozoic stands for “earlier life.” During the Proterozoic, the mantle cooled down, allowing the development of a proper thicker lithosphere (Palin et al., 2022; Stern, 2018). It is still debated when plate tectonics exactly started, but the emergent view is that some tectonics, i.e., movement and deformation of the planet’s surface, already existed in the Archean, but likely without a globally fragmented mobile lid. Such a mobile lid probably developed progressively during the Proterozoic, while subduction zones become more efficient in recycling the oceanic lithosphere. A fully developed plate mosaic was likely fully established 1000 Ma (Stern, 2018; Palin et al., 2020), with the operation

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of modern-day plate tectonics. It is also debated when the continents developed, with two prevailing perspectives. One suggests that they developed during the Proterozoic, while others think that this is related to a preservation bias and that the continental crust was already widespread by the end of the Archean (see Palin et al., 2020, for a discussion). Nevertheless, some form of Wilson and supercontinent cycles was already operating during the Proterozoic, with episodes of supercontinent breakup and reformation (see Chapter 3). The first supercontinent for which there is large evidence formed during the Paleoproterozoic 2100–1800 Ma. It is called Nuna (also known as Columbia) and resulted from a global scale collisional event between early continents (Rogers and Santosh, 2002). There is strong evidence that mountain building resulting from this collision started around 2000 Ma, and that the explosion of life might have played an important role in it (Parnell and Brolly, 2021). Nuna broke up in the Mesoproterozoic, about 1400 Ga. During the Neoproterozoic two other supercontinents formed: Rodinia 1000 Ma and Pannotia 600 Ma (e.g., Scotese, 2009). A Great Oxidation Event occurred in the Paleproterozoic, 2400–2200 Ma, related to the rapid increase of free oxygen in the atmosphere produced by cyanobacteria, which in turn caused the extinction of many anaerobic species. This event left a geological record of oxides of iron deposits, known as BIFs (banded iron formations). This, together with the changes in the modes of plate tectonics, caused important climate changes (see, e.g., Levin and King, 2016). The first glaciation is thought to have occurred in the beginning of the Paleoproterozoic, around 2300 Ma, and there is strong evidence that three major glaciations occurred in the Neoproterozoic. Two of these glaciations, the Sturtian and Marinoan, were near global and may have caused the Earth to enter a Snowball state (Hoffman et al., 2017). During this large period of the Earth’s history, and because of the Wilson and the supercontinent cycle, the ocean dynamics likely underwent many changes. It is thought that the supercontinent Nuna was vastly covered by shallow epicontinental seas with large shelf areas. However, this may not have been the case during the tenure of Rodinia, as this was a polar supercontinent centered around the South Pole, which may have led to the formation of large glaciated continental regions. We can speculate that in between the tenure of supercontinents ocean circulation may have resembled the one we see on Earth today but with variations largely controlled by the disposition of the continents and ocean seaways.

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The earlier Proterozoic life was not that different from the Archean, with blue-green cyanobacteria forming algal mats and widespread anaerobic prokaryotes. Stromatolites expanded worldwide and declined by the end of the Proterozoic. The appearance of advanced single-celled, eukaryotes and multicellular life coincides with the increase of free oxygen in the atmosphere, between 2100 and 1600 Ma. The earliest evidence of eukaryotes (complex cells with organelles) dates from 1850 Ma (Knoll et al., 2006), and multicellular organisms were already present by 1700 Ma (Bonner, 1998). Algae-like multicellular land plants are dated back to 1000 Ma (Strother et al., 2011). The Ediacaran biota, which includes the first complex life forms, appeared during an interval known as the Ediacaran period, between 635 and 541 Ma, marking the end of the Proterozoic ( Jun-Yuan et al., 2000).

4.4 Phanerozoic Eon (541–0 Ma) The Phanerozoic is the eon we are living in. It started 541 Ma ago and it is generally divided into three eras: Paleozoic, Mesozoic, and Cenozoic. During the Phanerozoic, modern plate tectonics was operative and the Earth’s climate stabilized, allowing complex life to diversify and evolve. Phanerozoic stands for “visible life.” After the break-up of Rodinia, 750 Ma, the continents moved apart and briefly came together at 600 Ma to form a short-lived supercontinent, Pannotia (Donnadieu et al., 2004). At the start of the Phanerozoic, 550 Ma, a mega-continent named Gondwana was formed. Gondwana later collided with North America and Europe to form the latest supercontinent on Earth, Pangaea (which stands for “all land”). Pangaea gathered toward the end of the Paleozoic, 335 Ma, and had a tenure of about 130 Myrs. Simultaneously, a global super ocean formed called Panthalassa, which means “all sea.” Pangaea started to break apart around 200 Ma, well within the Mesozoic era (Rogers and Santosh, 2004), forming the Atlantic Ocean. What remains of the Panthalassa is now called the Pacific, and while the ocean basin is still partially the same, the Panthalassa seafloor was almost fully recycled at its peri-oceanic subduction zones (the famous ring of fire discussed in Chapter 3). There were a few significant climate variations during the Phanerozoic, with several periods of greenhouse and icehouse. The Pangea climate was hotter than today, in part because its interior was deserted and dry, even when internal shallow seas started to invade the supercontinent during the Mesozoic (e.g., Parrish, 1993). After its breakup and dispersal, ocean

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and climate circulation become closer to the present day. Notwithstanding, several events caused major changes, including the closing of the Panama gateway, allowing for the formation of the Gulf Stream, the separation of Australia from Antarctica, forming the Antarctic Circumpolar Current, and the formation of the Himalaya, as the result of the collision between India and Eurasia, which caused the Indian Monsoon. A very warm period occurred in the Early Eocene, right after the Paleocene–Eocene Thermal Maximum 55.5 Ma. Since then, the climate has seen a gradually cooling, and the last 2 Myrs have been characterized by a series of glaciations and interglacial periods. The Phanerozoic is characterized by massive changes in life. The Cambrian explosion, 541 Ma, corresponded to the appearance of a great abundance of complex and diversified invertebrates and vertebrates. Most of the known animal phyla appeared during this period (Budd, 2003). Around 420 Ma, plants colonized land, eventually forming vast rainforests. Arthropods also made their transition to land and vertebrates developed lungs and legs. Until the mid-Carboniferous (320 Ma), the tetrapods were mostly dominated by amphibians, after which several climate and oceanographic events led to a loss of their habitat, opening the opportunity for reptiles to develop. The Mesozoic (251–66 Ma) land was largely dominated by dinosaurs, while conifers dominated the terrestrial flora. During the Cretaceous, plants with flowers appeared. By the end of the Cretaceous, most of the nonavian dinosaurs become extinct. The first birds and mammals also evolved during the Mesozoic, but they only managed to succeed during the following era, the Cenozoic. The formation of Pangea and its subsequent break-up likely contributed to the large diversification of life, destroying and creating new niches. However, life’s evolution does not come as a simple and linear progression. The Phanerozoic was also marked by five mass extinction events that could have annihilated life on Earth (Raup and Sepkoski, 1982). It is hard to tell from the geological record, but there were likely other extinction events before the Phanerozoic, during the Great Oxidation Event and the Snowball period. Some authors suggest that we are undergoing a sixth Phanerozoic mass extinction caused by human activity.

5. Final remarks The present book is about ocean tides throughout the Earth’s history. However, while we can observe and measure tides on the present-day Earth and even collect historical registers, it is not possible for the immensity of

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geological time. For that, we have to use modeling tools. Notwithstanding, we can use the geological record in our favor. There is a special type of sedimentary rock, tidalites, that register past tidal cycles and can provide crucial information about past tidal parameters. They can also be used as proxies to validate numerical models of deep time tides. The links between tectonics, tides, and past geological record have made great progress in the last years, but, in many ways, it is just starting. This book will certainly be a foundational initiative for the exciting work we still have ahead.

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Parnell, J., Brolly, C., 2021. Increased biomass and carbon burial 2 billion years ago triggered mountain building. Commun Earth Environ 2, 238. https://doi.org/10.1038/s43247021-00313-5. Parrish, J.T., 1993. Climate of the supercontinent Pangea. J. Geol. 10, 215–233. Raup, D., Sepkoski Jr., J., 1982. Mass extinctions in the marine fossil record. Science 215 (4539), 1501–1503. Rogers, J.J., 1996. A history of continents in the past three billion years. J. Geol. 104 (1), 91–107. https://doi.org/10.1086/629803. Rogers, J.J., Santosh, M., 2002. Configuration of Columbia, a Mesoproterozoic supercontinent. Gondw. Res. 5 (1), 5–22. https://doi.org/10.1016/S1342-937X(05)70883-2. Rogers, J.J.W., Santosh, M., 2004. Continents and Supercontinents. Oxford University Press. Scotese, C.R., 2009. Late Proterozoic plate tectonics and palaeogeography: a tale of two supercontinents, Rodinia and Pannotia. Geol. Soc. Lond. Spec. Publ. 326, 67–83. https://doi.org/10.1144/SP326. Silver, P.G., Behn, M.D., 2008. Intermittent plate tectonics? Science 319, 85–88. Stanley, S.M., 1999. Earth System History. W.H. Freeman and Company, New York, ISBN: 978-0716728825. Stern, R.J., 2018. The evolution of plate tectonics. Phil. Trans. R. Soc. A 376, 20170406. https://doi.org/10.1098/rsta.2017.0406. Strother, P.K., Battison, L., Brasier, M.D., et al., 2011. Earth’s earliest non-marine eukaryotes. Nature 473 (7348), 505–509. https://doi.org/10.1038/nature09943. Wilde, S.A., Valley, J.W., Peck, W.H., Graham, C.M., 2001. Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature 409 (6817), 175–178. https://doi.org/10.1038/35051550. PMID 11196637. S2CID 4319774. Yu, G., Jacobsen, S.B., 2011. Fast accretion of the earth with a late moon-forming giant impact. Proc. Natl. Acad. Sci. U. S. A. 108, 17604.

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CHAPTER 6

Hadean and Archean (4600–2500 Ma) Hannah S. Daviesa, João C. Duarteb, and Mattias Greenc a

Helmholtz Center Potsdam, GFZ German Research Center for Geosciences, Potsdam, Germany Geology Department and Instituto Dom Luiz (IDL), Faculty of Sciences of the University of Lisbon, Lisbon, Portugal c School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom b

1. Introduction The Hadean (4600–4000 Ma) and Archean (4000–2500 Ma) Earth would have been unrecognizable to us. The surface would have been either magma or ocean (with continents emerging towards later stages), or pockmarked and scarred by impacts and intense volcanism, and the atmosphere unbreathable due to a lack of free oxygen (Catling and Zahnle, 2020). Depending on when exactly you chose to view the young Earth, oceans may or may not have condensed (Wilde et al., 2001). If they had, they would dominate and heavily contrast the sparse brown-gray land, devoid of life until much later in Earth’s history. Life would soon emerge in the oceans; however, it is not known when and where exactly it first developed, but it could have been as early as the beginning of the Archean, 4000 Ma ( Javaux, 2019). There is not much we know about the early Earth because only tiny fragments of geological evidence from the period remain. However, we do know how the Earth and Moon formed and roughly when the oceans condensed. The Earth formed from agglomeration of planetesimals in the solar nebula (Wetherill, 1990) around 4600 Ma. Around 4450 Ma, Earth and a Mars sized protoplanet called Theia collided (Canup and Asphaug, 2001). The ejecta from the impact, a mix of material from the two bodies, accreted to form the Moon. It would have formed very close to the Earth, likely near the Roche limit – the closest orbit an object can be to a planet before being torn apart by tidal forces – around 20,000 km away (Darwin, 1899). Due to their close proximity to each other, both the Earth and Moon would have experienced extremely large tidal forces, leading to a very rapid lunar recession and slowdown of Earth’s spin. In just a couple of hundred million years, the Moon would have reached a distance of at least 200,000 km – 10 times the distance at which it formed. A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00020-0

Copyright © 2023 Elsevier Inc. All rights reserved.

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This initial period of Lunar recession was driven by tidal forces within the molten rock of Earth and the Moon, either in magma oceans or as a solid Earth tide. However, the oceans soon condensed: Wilde et al. (2001) found zircon evidence for liquid water on Earth at 4400 Ma in the Jack Hills formation in Western Australia (see also Compston and Pidgeon, 1986), which suggests relatively rapid condensation of the atmosphere following the Moon forming impact. The Earth would then look a little more recognizable with a large area of ocean covering probably around 85% the Earth’s surface, compared to 71% at present (Korenaga et al., 2017). The Earth may have in fact resembled a “wet Venus” (Fig. 6.1B and C), with small outcrops of land surrounded by large oceans, likely shallower than present day but perhaps containing as much water by volume.

Fig. 6.1 (A) NOAA ETOPO1 bathymetry used as the base map for the present-day Earth collection of tidal simulations. Present-day Venus topography from Green et al. (2019) “flooded” with an ocean of an average depth of 750 m (B) and 4000 m (C) to resemble the Archean Earth. Two selected maps (D,E) from the Archean ensemble of randomized bathymetries to illustrate what a typical “Archean map” from the ensemble looks like. Note that the color scale saturates at 6000 m.

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In this chapter, we will explore how large the tides in an Archean Ocean may have been to constrain the lunar recession rate and associated change in daylength. Because of the lack of knowledge of what Archean Earth looked like, we will take an ensemble approach and discuss results from a suite of simulations using the semirandom bathymetries presented by Blackledge et al. (2020).

2. Methods 2.1 Tidal modeling The Archean tides were simulated using the portable Oregon State University Tidal Inversion Software (OTIS). OTIS has been used extensively to simulate deep-time, present day, and future tides (Byrne et al., 2020; Egbert et al., 2004; Green, 2010; Green et al., 2018, 2020; Wilmes and Green, 2014; Wilmes et al., 2017), and it was benchmarked against other forward tidal models and shown to perform well (Stammer et al., 2014). The model numerically solves the linearized shallow water equations (Egbert et al., 2004):
∂U (6.1) + f
U ¼ gHr η  ηSAL  ηEQ  F ∂t ∂η  r∙U ¼ 0 (6.2) ∂t Here, U ¼ uH is the tidal volume transport (u is the horizontal velocity vector and H is the water depth), f is the Coriolis parameter, g is acceleration due to gravity, η is the sea-surface elevation, ηSAL is the self-attraction and loading elevation, η EQ is the elevation of the equilibrium tide, and F is the tidal energy dissipation term. The latter has two components: F ¼ FB + FW. FB parameterizes bed friction and FW represents energy losses due to tidal conversion, that is, the generation of a baroclinic tide. The first term is represented using the standard quadratic law, FB ¼ CDu j u j, where CD ¼ 0.003 is a dimensionless drag coefficient (Taylor, 1920). The tidal conversion term is given by FW ¼ CU, where the conversion coefficient, C, is given by (Green and Huber, 2013; Green and Nycander, 2013; Zaron and Egbert, 2006) C ðx, yÞ ¼ γ

NHN ðrH Þ2 8πω

(6.3)

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Here, γ ¼ 50 represents a dimensionless scaling factor representing unresolved bathymetric roughness, NH is the buoyancy frequency at the seabed, N represents the vertical average of the buoyancy frequency, and ω is the frequency of the tide. The buoyancy frequency, N, is given by N2 ¼  g/ρ ∂ ρ/∂z, where ρ is the density. The density stratification is uncertain for most ancient oceans, so we used values of N based on a statistical fit to observed present-day values (Zaron and Egbert, 2006): N(x,y) ¼ 0.00524exp(z/1300), where z is the vertical coordinate. Each run simulated the principal lunar (M2) tidal constituent only. The model output consists of amplitudes and phases of the sea surface elevations and transports. These were used to compute the tidal dissipation rate, D, as (see Egbert and Ray, 2001, for details): D¼WrP

(6.4)

The work and flux are given by (using angular brackets to denote averages over a tidal cycle)


 W ¼ gρ U  r ηEQ + ηSAL (6.5) and P ¼ gρhUηi

(6.6)

We chose orbital parameters representing an Early Archean Earth and a 13.1 h day, leading to a 6.7 h M2 tidal period. Consequently, the Lunar distance was 266,000 km (Waltham, 2015), giving an equilibrium forcing a factor 3.4 larger than at present.

2.2 Bathymetry We used three collections of bathymetries. The first set of simulations used a present-day bathymetry (NOAA ETOPO1 – see https://data.nodc.noaa. gov/cgi-bin/iso?id¼gov.noaa.ngdc.mgg.dem:316# for details; Fig. 6.1A). Simulations with the present-day bathymetry were done using both present-day and Archean forcing, and both forcings were used in the next set of bathymetries as well. These used a Venusian topography (Green et al., 2019; our Figs. 6.1B and C), with two water depths: “intermediate,” which used a 750-m-deep ocean on average, and “deep” had a 4000 m deep ocean. The third set is the ensemble of randomized bathymetries presented by Blackledge et al. (2020), which have been modified for this work. They were flooded until the land area averaged 5%–15% while retaining present-day ocean volume by raising the seafloor where needed (Fig. 6.1D and E). As these are randomly generated maps, they do not

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contain underwater features such as continental shelves, ridges, or subduction zones (which may be representative of an Archean Earth before plate tectonics started). We also ran a water world control with a global ocean 4 km deep to compare the modeled results with.

3. Results 3.1 Present-day Earth bathymetry The dissipation rate for the Archean tidal forcing in the present day is 15 TW or more than 5 times larger than the present-day modeled value of 2.9 TW (Fig. 6.2B). Because of the different periods for the M2 tide between the simulations, and that peaks in dissipation comes from tidal resonances, the locations of peak tidal dissipation are different. For example, the presentday North Atlantic is near resonant (Fig. 6.2A), but with Archean forcing, the South Atlantic has a stronger tidal signal. This makes sense from a dynamic perspective because the South Atlantic is narrower and hence resonant for shorter periods (see, e.g., Arbic and Garrett, 2010 and Chapter 4).

3.2 Venusian topography The simulations with a Venusian topography on Earth also show increased tidal energetics for the Archean tidal parameters compared to present-day forcing (1.2–2.0 TW dissipation for present day vs. 3.1–6.6 TW for the Archean forcing with deep and intermediate oceans, respectively, see Fig. 6.1B and C). Interestingly, the present-day dissipation rates increase as the ocean deepens, whereas for the Archean set up, the intermediate depth is more than twice as energetic (6.6 TW) as the deep case (3.1 TW). There is thus an order of magnitude difference in the dissipation rates for a shallow Venusian bathymetry on Earth due to the enhanced forcing. The reason the deeper runs are more dissipative than the shallow one is simply because there is more ocean surface area to dissipate energy in the deeper simulations.

3.3 Archean ensemble In the ensemble simulations with a randomized bathymetry, we also find an enhanced tidal signal for the Archean with average global M2 tidal dissipation and amplitudes being around 167% higher than the rates using present-day forcing (5.0 TW average for the Archean forcing against 3.0 for present day; Fig. 6.1E). Most of the tidal dissipation occurs below 1000 m – here defined as “deep” dissipation. In the median result, deep dissipation made up over 80% of the total value – this is representative for most of the ensemble.

Fig. 6.2 M2 tidal amplitudes for present-day bathymetry with (A) present-day tidal forcing, (B) Archean tidal forcing, Venus with Archean tidal parameters, and two different ocean setups – (C) intermediate 750 m avg. depth and (D) deep 4000 m avg. depth. Two maps from the Archean ensemble with the median (E) and maximum (F) tidal dissipation values. The scale bar saturates at 2 m.

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4. Discussion The M2 equilibrium forcing is 3.4 times larger in our Archean setup than present day. Consequently, our Archean simulations are usually more energetic than the ones with present-day forcing. The simulations with a present-day Earth bathymetry experience an enhanced dissipation with a factor 5 due to the newly developed South Atlantic resonance, whereas in the simulations using a Venusian topography on Earth, there is a factor 1.5–10 difference depending on depth. The ensemble simulations, on the other hand, have an average dissipation 1.7 times higher with the Archean forcing and the peak ensemble simulations, which dissipates 9.2 TW under Archean conditions (Fig. 6.1F) and dissipates 2.2 TW for present-day forcing. The water world dissipation was around 0.04 TW for the Archean, meaning that there is a strong modulation of the tide by even the sparse land. Deep dissipation rates show that even without major ocean basin features such as mid ocean ridges there is still a large portion of the tide dissipating along the ocean floor. Our Archean ensemble and Venusian bathymetry simulations average around 4 TW. This matches Daher et al. (2021)’s results for 3900 Ma. Daher et al. (2021) use their tidal dissipation results in an orbital evolution model and interestingly fail to reconstruct an Earth–Moon collision at 4500 Ma. This suggests that Archean dissipation rates of 4 TW are underestimated, or that the tides between 4500 and 3900 Ma must have been far larger than 4 TW (or both). This could be accounted for by better representing other tidal processes, for example, the solid earth tide may be fundamental in contributing to the tidal acceleration that occurred during the first hundred million years or so after the Moon forming impact. Future work is therefore required to tidally reconstruct the Moon forming impact. Furthermore, Lathe (2004) argues that the potentially enhanced tidal energy in the early Archean (around 3900 Ma) led to fast tidal cycling of the potentially vast intertidal zone in the early Archean (3900 Ma). This is one mechanism by which early self-replicating molecules, such as RNA and DNA, may have developed, giving further motivations for studies of Archean tides.

References Arbic, B.K., Garrett, C., 2010. A coupled oscillator model of shelf and ocean tides. Cont. Shelf Res. 30 (6), 564–574. https://doi.org/10.1016/j.csr.2009.07.008. Blackledge, B.W., et al., 2020. Tides on other earths: implications for exoplanet and palaeotidal simulations. Geophys. Res. Lett. 47 (12), e2019GL085746. https://doi.org/ 10.1029/2019GL085746.

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Byrne, H.M., et al., 2020. Tides: a key environmental driver of osteichthyan evolution and the fish-tetrapod transition? Proc. R. Soc. A Math. Phys. Eng. Sci. 476 (2242), 20200355. https://doi.org/10.1098/rspa.2020.0355. Royal Society. Canup, R.M., Asphaug, E., 2001. Origin of the moon in a giant impact near the end of the Earth’s formation. Nature 412 (6848), 708–712. https://doi.org/10.1038/35089010. Catling, D.C., Zahnle, K.J., 2020. The Archean atmosphere. Sci. Adv. 6 (9), eaax1420. https://doi.org/10.1126/sciadv.aax1420. American Association for the Advancement of Science. Compston, W., Pidgeon, R.T., 1986. Jack Hills, evidence of more very old detrital zircons in Western Australia. Nature 321 (6072), 766–769. https://doi.org/10.1038/321766a0. Daher, H., et al., 2021. Long-term earth-moon evolution with high-level orbit and ocean tide models. J. Geophys. Res. Planets 126 (12), e2021JE006875. https://doi.org/ 10.1029/2021JE006875. Darwin, G.H., 1899. The Tides and Kindred Phenomena in the Solar System: The Substance of Lectures Delivered in 1897 at the Lowell Institute. Houghton, Mifflin and Company, Boston, Mass. Egbert, G.D., Ray, R.D., 2001. Estimates of M2 tidal energy dissipation from TOPEX/ Poseidon altimeter data. J. Geophys. Res. Oceans 106 (C10), 22475–22502. https:// doi.org/10.1029/2000JC000699. Egbert, G.D., Ray, R.D., Bills, B.G., 2004. Numerical modeling of the global semidiurnal tide in the present day and in the last glacial maximum. J. Geophys. Res. Oceans 109 (C3). https://doi.org/10.1029/2003JC001973. John Wiley & Sons, Ltd. Green, J.A.M., 2010. Ocean tides and resonance. Ocean Dyn. 60 (5), 1243–1253. https:// doi.org/10.1007/s10236-010-0331-1. Green, J.A.M., Huber, M., 2013. Tidal dissipation in the early Eocene and implications for ocean mixing. Geophys. Res. Lett. 40 (11), 2707–2713. https://doi.org/10.1002/ grl.50510. Green, J.A.M., Nycander, J., 2013. A comparison of tidal conversion parameterizations for Tidal models. J. Phys. Oceanogr. 43 (1), 104–119. https://doi.org/10.1175/JPO-D12-023.1. Boston MA, USA: American Meteorological Society. Green, J.A.M., et al., 2018. Is there a tectonically driven supertidal cycle? J. Geophys. Res. Oceans 45 (8), 3568–3576. https://doi.org/10.1002/2017GL076695. John Wiley & Sons, Ltd. Green, J.A.M., Way, M.J., Barnes, R., 2019. Consequences of tidal dissipation in a putative venusian ocean. Astrophys. J. Lett. 876 (2), L22. https://doi.org/10.3847/2041-8213/ ab133b. American Astronomical Society. Green, J.A.M., et al., 2020. Weak tides during Cryogenian glaciations. Nat. Commun. 11 (1), 6227. https://doi.org/10.1038/s41467-020-20008-3. Javaux, E.J., 2019. Challenges in evidencing the earliest traces of life. Nature 572 (7770), 451–460. https://doi.org/10.1038/s41586-019-1436-4. Korenaga, J., Planavsky, N.J., Evans, D.A.D., 2017. Global water cycle and the coevolution of the Earth’s interior and surface environment. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 375 (2094), 20150393. https://doi.org/10.1098/rsta.2015.0393. Lathe, R., 2004. Fast tidal cycling and the origin of life. Icarus 168 (1), 18–22. https://doi. org/10.1016/j.icarus.2003.10.018. Stammer, D., et al., 2014. Accuracy assessment of global barotropic ocean tide models. Rev. Geophys. 52 (3), 243–282. https://doi.org/10.1002/2014RG000450. Taylor, G.I., 1920. Tidal friction in the Irish Sea. Proc. R. Soc. London, Ser. A 96, 1–33. Waltham, D., 2015. Milankovitch period uncertainties and their impact on cyclostratigraphy. J. Sediment. Res. 85 (8), 990–998. https://doi.org/10.2110/jsr.2015.66. Wetherill, G.W., 1990. Formation of the Earth. Annu. Rev. Earth Planet. Sci. 18 (1), 205–256. https://doi.org/10.1146/annurev.ea.18.050190.001225. Annual Reviews.

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Wilde, S.A., et al., 2001. Evidence from detrital zircons for the existence of continental crust and oceans on the Earth 4.4 Gyr ago. Nature 409 (6817), 175–178. https://doi.org/ 10.1038/35051550. Wilmes, S.-B., Green, J.A.M., 2014. The evolution of tides and tidal dissipation over the past 21,000 years. J. Geophys. Res. Oceans 119 (7), 4083–4100. https://doi.org/ 10.1002/2013JC009605. Wilmes, S.-.B., Mattias Green, J.A., Gomez, N., Rippeth, T.P., Lau, H., 2017. Global tidal impacts of large-scale ice sheet collapses. J. Geophys. Res. Oceans 122 (11), 8354–8370. https://doi.org/10.1002/2017JC013109. Zaron, E.D., Egbert, G.D., 2006. Estimating open-ocean Barotropic Tidal dissipation: the Hawaiian Ridge. J. Phys. Oceanogr. 36 (6), 1019–1035. https://doi.org/10.1175/ JPO2878.1. Boston MA, USA: American Meteorological Society.

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CHAPTER 7

Proterozoic (2500–541 Ma) Mattias Greena, Christopher Scoteseb, and Hannah S. Daviesc a

School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Earth and Planetary Sciences, Northwestern University, Evanston, IL, United States Helmholtz Center Potsdam, GFZ German Research Center for Geosciences, Potsdam, Germany

b c

1. Introduction The Proterozoic eon spans 2500–541 Ma and thus covers over 40% of Earth’s history. It saw major Earth system events (Stanley and Luczaj, 2014), such as the oxygenation of the atmosphere between 2400 and 2000 Ma, which in turn kickstarted the evolution of complex life (El Albani et al., 2010; Stanley and Luczaj, 2014). This was likely a result of the tides which, due to tidal drag, reduced daylength enough to allow photosynthesis to flourish (Daher et al., 2021; Klatt et al., 2021). Later on in the eon, during the Cryogenian (715–635 Ma), Earth was probably home to a series of prolonged and severe glaciations (Hoffman et al., 2017; Pasˇava et al., 2019), dubbing the period “Snowball Earth.” Such severe glaciations led to a stable climate state, and there has been some debate over how Earth escaped the Snowball and how severe the glaciations may have been (Hoffman et al., 2017; Pierrehumbert et al., 2011; Williams et al., 2016). It has been speculated that maybe the tides were a contributor to the escape from the icehouse by driving an enhanced ice melt (Wunsch, 2016), but this was not the case because Cryogenian tides were very weak (Green et al., 2020). The friction induced between the tides and Earth slows the rotational period, making days longer (Bills and Ray, 1999; Daher et al., 2021; Darwin, 1898; MacDonald, 1964). Consequently, to conserve angular momentum, which is driven by torques that are exerted on the body, the Moon must recede. The rotational evolution depends on the amount of tidal energy dissipated in the ocean, and tides thus exert a first-order control on the orbital evolution of the Earth-Moon system (Bills and Ray, 1999; Munk, 1968). A problem then arises: the current lunar recession rate is anomalously high at 3.8 cm year
1 (Dickey et al., 1994) and if one makes the erroneous assumption that this rate has been constant through long periods of time, the Moon cannot be more than 1500 Myr old (this is sometimes seen as a A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00004-2

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capture rather than impact event referred to as the Gerstenkorn event – see Munk, 1968 for a discussion). This is in conflict with the radiometric age of the Moon, which estimates that the Moon is 4500 Myr old (Barnes, 2017; Dickey et al., 1994), suggesting that at least one of these age estimates must be wrong. The discrepancy between the different lunar age estimates is the result of an inconsistent rate of tidal dissipation during Earth’s history. This is because the movement and emergence of continents change the size and shape of ocean basins, which in turn, act to change the tidal amplitudes, and hence the dissipation rates, through resonance effects (Arbic et al., 2009; Green, 2010; Green et al., 2017). The current continental configuration, along with the size and shape of present-day ocean basins means that we are currently in a tidal maximum (Green et al., 2018), but where we are in the maximum is not clear. Determining the tidal dissipation rates for Earth’s deep past is an important factor for a range of Earth’s system processes beyond orbital evolution. For example, tides control the locations of productive shelf sea fronts by driving vertical mixing that outcompetes buoyancy inputs at the sea surface (Simpson and Hunter, 1974) and tidally driven mixing sustains primary production in the ocean by driving vertical fluxes of nutrients (Sharples et al., 2007; Tuerena et al., 2019). Furthermore, tides provide some of the energy needed to sustain the climate-regulating global overturning circulation by balancing deep-water formation (Wilmes et al., 2021; Wunsch and Ferrari, 2004). However, little is known about the surface of the Earth during the earliest part of the Paleoproterozoic Era (2500–1500 Ma), and any tidal simulations of that era would have to be done from a statistical perspective (see, e.g., Blackledge et al., 2020). This period is therefore not described in detail here, as the Archaean simulations presented in Chapter 6 can be extended to cover the first 1000 Myr of the Proterozoic (i.e., 2500–1500 Ma). The period spanning 1800–800 Ma is known as the “Boring Billion,” because of its tectonic, evolutionary, and (warm) climatic stability (Cawood et al., 2018; Liu et al., 2019; Young, 2013), and we can simulate the tides on Earth from around 1500 Ma using the reconstructions in Scotese (2009). Approximate as these may be, they do provide large-scale tectonic structures and enable direct simulations, albeit ones that need to be interpreted with some caution; were the tides boring as well? The supercontinent Rodinia formed around 1100–1000 Ma and broke up during the Cryogenian, scattered, and then briefly reformed as Pannotia towards the end of the Cryogenian; the breakup was complete around 633 Ma (Li et al., 2008; Scotese, 2009). It has been suggested that the tides are weak during a

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supercontinent stage but that they reach tidal maxima around 200 Myr before a supercontinent assembles (Green et al., 2018), as is the case of today’s oceans (Davies et al., 2020; Green et al., 2018). If this supertidal cycle is a consistent feature of the ocean, we would expect the early Mesoproterozoic (1600–1000 Ma, although our simulations start at 1500 Ma) to be tidally energetic, and the prolonged period of Rodinia more quiescent (as suggested by Green et al., 2020). It has been proposed that Earth hosted two major glacial events during the Cryogenian: the Sturtian glaciation, between 717 and 660 Ma, and the Marinoan glaciation between 650 and 635 Ma (Hoffman et al., 2017; Rooney et al., 2015). Both saw continental ice advance down to within 12° of the equator (Macdonald et al., 2010; Pierrehumbert et al., 2011), possibly leaving an open equatorial ocean. This latter state, with an ice-free ocean, is referred to as “Slushball Earth” (Allen and Etienne, 2008) and is the one included here. The tides during the Cryogenian were extensively investigated by Green et al. (2020), and are discussed briefly here for completeness. After the Cryogenian, Earth started to change as we entered the Ediacaran (635–541 Ma). Life went from being dominated by microbes to one rich in animals, the climate warmed from the Cryogenian icehouse, and the continents started to scatter as Rodinia/Pannotia broke up. These large changes in Earth’s tectonic configuration and the potential generation of new shallow shelf seas have not been explored before. Consequently, in this chapter, we present new results for the period between 1500 and 800 Ma; these are then merged with the existing simulations for the Cryogenian (Green et al., 2020), covering 750–600 Ma, before we close the chapter with new simulations for the Ediacaran until we reach the ProterozoicCambrian Boundary at 540 Ma.

2. Methods 2.1 Tidal modeling The Proterozoic tides were simulated using the Oregon State University Tidal Inversion Software (OTIS). OTIS has been used extensively to simulate deeptime, present day, and future tides (Egbert et al., 2004; Green et al., 2017, 2020; Green, 2018; Wilmes and Green, 2014), and it was benchmarked against other forward tidal models and shown to perform well (Stammer et al., 2014). The model numerically solves the linearized shallow water:
∂U + f × U ¼ gHr η
ηSAL
ηEQ
F ∂t

(7.1)

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∂η
rU¼0 (7.2) ∂t Here, U ¼ uH is the tidal volume transport (u is the horizontal velocity vector and H is the water depth), f is the Coriolis parameter, g is acceleration due to gravity, η is the sea-surface elevation, ηSAL is the self-attraction and loading elevation, ηEQ is the elevation of the equilibrium tide, and F the tidal energy dissipation term. The latter has two components, so that F ¼ FB + FW. FB parameterizes bed friction and FW represents energy losses due to tidal conversion, i.e., the generation of a baroclinic tide. The first term is represented using the standard quadratic law, FB ¼ CD u ju j, where CD ¼ 0.003 is a dimensionless drag coefficient. The tidal conversion term is given by FW ¼ CU, with a conversion coefficient, C, given by (Green and Huber, 2013; Green and Nycander, 2013; Zaron and Egbert, 2006) NHN (7.3) ðrH Þ2 8πω Here, γ ¼ 50 represents a dimensionless scaling factor representing unresolved bathymetric roughness, NH is the buoyancy frequency at the seabed, N represents the vertical average of the buoyancy frequency, and ω is the frequency of the tide. The buoyancy frequency, N, is given by N2 ¼
g/ρ ∂ ρ/∂z, where ρ is the density. The density stratification is uncertain for most ancient oceans, so we used values of N based on a statistical fit to observed present-day values (Zaron and Egbert, 2006): N(x,y) ¼ 0.00524exp(
z/1300), where z is the vertical coordinate, and the constants 0.00524 and 1300 have units of s
1 and m, respectively. We tested the sensitivity to our choice of γ by changing it to 25 or 100 and found that this only changes the dissipation rates by less than 5% globally. Consequently, we only discuss simulations with γ ¼ 50 for nonglaciated time slices. However, for the Cryogenian time slices where the Slushball may have been present, we assumed that the stratification was weaker (Ashkenazy et al., 2013, 2014) and used a reduced gamma (γ ¼ 25) and enhanced bed friction (Cd ¼ 0.006) to account for an enhanced drag under the ice (see Green et al., 2020 for details about the Cryogenian set up). C ðx, yÞ ¼ γ

2.2 Bathymetry The paleo-bathymetries were created from the raster maps of Scotese (2009), which were imported into the GPlates software (Qin et al., 2012). The reconstructions were available at a time resolution of 50 to

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Fig. 7.1 Proterozoic bathymetries. Bathymetric maps for the time slices discussed (data from Scotese, 2009). Note that land is white and the depth saturates at 3000 m so trenches are not visible and that the abyssal ocean has a constant depth away from the ridges and trenches, hence the dark appearance. The lines in the abyssal ocean in the lower panels mark mid-oceanic ridges.

100 Myr depending on period; the new simulations for the Proterozoic presented were generated using the same intervals, whereas the Cryogenian simulations were originally generated at 10 Myr intervals (Green et al., 2020). The GPlates images were turned into ocean bathymetries by setting continental shelf seas to 200 m depth, and subduction zones to 5900 m. Midoceanic ridges were 2500 m deep at the crest and sloped linearly into the abyss over 5° in width. The abyssal plains were set to a depth that conserved present-day ocean volume once all the other bathymetric features were set. Example time slices are shown in Fig. 7.1. The glaciated time slices (715–660 Ma, and 650–635 Ma) then had a sealevel reduced by 500 m to within 10° of the equator. This was done to represent the reduced water column thickness under the thick sea ice that would have been present during Slushball Earth (Creveling and Mitrovica, 2014; Green et al., 2020).

2.3 Simulations and computations The daylength changed by several hours during the Proterozoic (Daher et al., 2021); a day length of 18.7 h has been proposed for 1400 Ma (Meyers and Malinverno, 2018), whereas the day length around 630 Ma has been suggested to be 21.9 h (Williams, 1997). For simplicity, we employ these two daylengths for the periods 1500–1000 Ma and 1000–540 Ma, respectively. Consequently, 1000 Ma was simulated with both 18.7- and

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Table 7.1 Summary of the orbital parameters used in the simulations leading to the M2 tidal amplitudes shown. Age [Ma]

Lunar distance [km]

Day length [hours]

M2 period [hours]

M2 amplification

1500–1000 1000–600 0

340,154 366,090 384,399

18.7 21.9 24

9.56 10.98 12.42

x1.44 x1.15 x1.00

21.9-h daylengths; furthermore, all time slices were simulated using present day forcing. The resulting tidal periods and tidal forcing are summarized in Table 7.1. Simulations for all time slices were done at 1/4° horizontal resolution in both latitude and longitude. This was achieved through linear interpolation from the original image data of the reconstructions which had a 1x1° resolution. Each simulation covered 20 days, of which 15 days were used for harmonic analysis of the tide. Simulations were done for the three dominant constituents, M2 (principal lunar), S2 (principal solar), and K1 (luni-solar declination). The model outputs amplitudes and phase lags for the surface elevation and transport vector for each simulated tidal constituent. We focus on the lunar semidiurnal tide, M2, in the following. The reason is that, at least for the present, S2 and K1 combined are less energetic than M2, and that changes in the M2 tide are usually mapped onto changes in S2 (see, e.g., Byrne et al., 2020 and Chapter 8). A more thorough discussion about the other constituents will be included in a future publication in preparation, which will also present results better constrained by tidal proxies; in the meantime, see Daher et al., 2021 for some discussions. The output was used to compute tidal dissipation rates, D, as the difference between the time average of the work done by the tide generating force (W) and the divergence of the horizontal energy flux (P) (Egbert and Ray, 2001): D¼W
rP

(7.4)

where W and P are given by


 W ¼ gρ U  r ηEQ + ηSAL

(7.5)

P ¼ gρhUηi

(7.6)

and In Eqs. (7.5)–(7.6) the angular brackets mark time-averages over a tidal period.

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3. Results 3.1 Present-day validation A present-day control simulation using the bathymetry from Scotese and Wright (2018) gives a root-mean-square error of 15 cm for the global M2 tidal amplitudes compared to the altimetry-constrained TPXO9 model (see https://www.earthbyte.org/paleodem-resource-scotese-and-wright2018/ and https://www.tpxo.net for details). A simulation with a reduced-detail present-day bathymetry, with a resolution similar to reconstructed bathymetries from 50 Ma, produced an uncertainty of about 20 cm (Green et al., 2017), while also implying an M2 dissipation rate that is some 75% higher than present-day simulations, due to a lack of bathymetric detail in the deep ocean. Consequently, our simulations using reconstructions probably overestimate the tidal dissipation rates (see also Blackledge et al., 2020), so in this chapter we normalize the dissipation rates with the 4.5 TW from the present-day degenerate bathymetry in Green et al. (2017) to allow for easier comparison to previous studies.

3.2 Tidal evolution 1500–750 Ma The early Mesoproterozoic, around 1500 Ma, has a quite energetic M2 tide (Figs. 7.2–7.4), particularly at 1250 Ma (see Figs. 7.2A, 7.3A, and 7.4). Using the deep-time topography but with present-day forcing (Figs. 7.3D and 7.4D), shows a reduced tide, both compared to present day and to the simulations with altered forcing. In the simulations with present day forcing, there is a maximum at 1150 Ma), This strongly implies that the amplification

Fig. 7.2 Proterozoic M2 amplitudes. Modeled M2 amplitudes, in meters, for the five selected time slices. The simulations from 750 and 635 Ma are discussed in detail in Green et al. (2020).

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Fig. 7.3 Proterozoic dissipation rates. As in Fig. 7.2 but showing tidal dissipation rates; note that the color scale saturates at 0.05 W m
2.

Fig. 7.4 Proterozoic integrated dissipation rates Globally integrated tidal dissipation rates, normalized by the 4.5 TW rate for the deconstructed present-day bathymetry used by Green et al. (2017). The solid line shows results using the appropriate tidal forcing (see Table 7.1), with stars (*) and plus signs (+) indicating the periods when the daylength was 18.7 or 21.9 h, respectively. The dash-dotted line shows results from simulations with present-day (PD) forcing; note that 1000 Ma was simulated using all three forcings. The gray block arrows show the presence of supercontinents, with P denoting Pannotia.

using the appropriate forcing for the time is indeed due to the different forcing. The tidal forcing is then 44% larger than at present for the period between 1500 and 1000 Ma, but the time slices other than 1250 Ma are less energetic than present day. We therefore conclude that the enhanced tide at 1250 Ma is due to the change in day length leading to a resonant ocean for the semidiurnal tide; the same is seen in the present-day simulation for 1150 Ma (see Fig. 7.4). The simulations for the 1000 Ma time slice using different forcing highlight this further: as the amplification factor drops, so do the tidal energy levels out (Fig. 7.4).

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3.3 Tidal evolution 750–540 Ma The severe glaciations during the Cryogenian period (720–635 Ma, here represented by the Slushball simulations from Green et al., 2020) led to a prolonged period of weak tides because the ocean surrounding the supercontinent Pannotia was too large to host tidal resonances. A resonance happens when the basin has a width that is (odd multiples of ) half of the tidal wavelength (see Arbic et al., 2009, and Chapters 4 and 6), and the Panthalassa Ocean was too large to fulfill this criterion. The tidal energy dissipation rates were only 10–50% of today’s rates during the glaciations, but reached values of 80% of present-day values during the interglacial periods (e.g., 655 Ma and at the end of the Marinoan, at 635 Ma). Note that the integrated dissipation rate almost doubles compared the nearest glaciated time slices in the interglacial time slice at 655 Ma (Fig. 7.4), but that the levels are still only some 25% of present day. The 635 Ma time slice is an interesting one: Green et al. (2020) show that the removal of the ice led to a rapid – within a few 10 kyr – increase of the tides and argue that the icesheets, regardless of extent, severely dampened the Cryogenian tides. Here, we show in our Fig. 7.4 that despite an increased tidal forcing, the increase in Earth’s rotation rate, and hence shorter tidal periods and larger forcing, during the Cryogenian was responsible for the reduced tides by not facilitating tidal resonances. This highlights the complex interaction between rotation rates and topography, as the 635 Ma ocean is weakly resonant for present day forcing but not for the realistic forcing for the time (see Table 7.1). As we move to the last 100 Myr of the Mesoproterozoic period, the tides with appropriate forcing for the time slices stay quiescent whereas the ones with present day forcing continue to be slightly more energetic (note that the simulations from 600 to 540 Ma presented here are new). When we say energetic, it must be noted that even 540 Ma, which is a comparatively energetic time slice with respect to the previous 400 Myr or so, has a dissipation rate of only 60% of present-day rates because of the supercontinent Pannotia which briefly assembled after Rodinia broke up (see Figs. 7.1 and 7.4).Also, there is no tidal maximum between the two supercontinents Rodinia and Pannotia because the continents didn’t move far enough between the supercontinent stages to change the ocean basin geometries enough to allow for resonances. Extending the simulations back to 1500 Ma provides further support for the supertidal cycle being linked to the supercontinent cycle as suggested by Green et al. (2018). This points to a tidal maximum around 1250 Ma – about

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150–250 Myr before the assembly of the supercontinent Rodinia (Li et al., 2008; Scotese, 2009). The tides then remain quite quiescent until the next tidal maximum, proposed to occur in the Silurian at 440 Ma (Byrne et al., 2020 and Chapter 8), some 200 Myr after the breakup of Rodinia/Pannotia and 130 Myr before the formation of the following supercontinent, Pangea. This map onto the current supercontinent cycle: today Earth is in a very tidally energetic state (Davies et al., 2019; Green et al., 2018), and we are some 180 Myr away from the breakup of Pangea and 200–250 Myr away from the formation of the next supercontinent. The results presented in this chapter thus confirm the supertidal cycle – now covering the tenure of three supercontinents – and further emphasize the importance of linking orbital forcing and topography to accurately simulate tides on Earth and other planets.

4. Summary Tides are a key energy source for the Earth system and the drag introduced by the tides leads to the slow recession of the Moon and associated changes in day length (see, e.g., Daher et al., 2021 for a comprehensive account). Here, we add new results covering 1500–800 Ma and 540 Ma, that lend further support to the idea that tides were weaker than present-day values for most of Earth’s history (Munk, 1968), which strengthens the arguments for the old Moon age-model. With the results from Chapter 8, alongside previous work by Green et al. (2017, 2018), Davies et al. (2019), and Byrne et al. (2020), we now have tidal model simulations spanning more than two supercontinent cycles – from 1500 Ma in the past to 250 Myr into the future. These results show that the idea of a linked supertidal cycle, that is out of phase with the supercontinent cycle, is indeed a feature of Earth’s tides. Tides are, to first order, controlled by continental configuration; one could say that geology is the key controller of tidal energetics. We also suggest that the details of daylength are a second-order controller on the dynamics of the Earth-Moon system and the tides. Yes, 18.7 h give a different result to present-day daylength, but we argue that using the correct daylength to within an hour or so makes little difference; see, e.g., the results for the 1000 Ma time slice here. Bathymetry may have a larger effect on tidal dynamics than daylength, but the two combine to complicate the picture, with local resonances appearing at different locations and in different time slices (see Figs. 7.3–7.4, 7.5).

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The main assumption and source of uncertainty in deep-time tides simulations is bathymetry, inferred from tectonic reconstructions. New geodynamic reconstructions, with improved age models for the crust, are now available (Merdith et al., 2017; Müller et al., 2018) and could be used for the next stage of deep-time tidal simulations. The geological record may also provide more proxies of daylength, and tidal amplitudes and currents, that can be used to validate the simulations further, as in Byrne et al. (2020), Green et al. (2020), and Daher et al. (2021).

Acknowledgments The authors thank Joao Duarte for his patience editing the chapter. Simulations were done on Supercomputing Wales (funded by HEFCW) and the technical support of Aaron Owen is greatly appreciated. The authors thank the DETEST laboratory group members for proofreading and Phil Woodworth for useful comments.

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CHAPTER 8

Phanerozoic (541 Ma-present day) Mattias Greena, David Hadley-Prycea, and Christopher Scoteseb a School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Earth and Planetary Sciences, Northwestern University, Evanston, IL, United States

b

1. Introduction The Phanerozoic is our current geologic eon and it is made up of the Paleozoic (541–252 Ma), Mesozoic (252–66 Ma), and Cenozoic (66 Ma-present day) eras. It started in the Cambrian period 541 Ma when animals started to produce hard shells which are persevered in the fossil record (Stanley and Luczaj, 2014). Throughout the eon, complex plants evolved and moved onto land. Fish, insects, and tetrapods also emerged (and in some cases moved onto land), as did the dinosaurs and mammals. The eon has five major extinction events linked to several processes, including a meteorite impact. The eon has been geologically active, seeing the continued break up of one supercontinent, Pannotia, the assembly (around 310 Ma) and breakup (around 180 Ma) of the latest supercontinent, Pangea, and the subsequent dispersal of the continents, leading to the view of Earth as we know it. It is important to understand how tides evolve over geological time because they exert a first-order control on the evolution of the Earth-Moon system due to tidal drag slowing down Earth’s spin and driving the Moon away from Earth as a result (Bills and Ray, 1999; Daher et al., 2021; MacDonald, 1964; Munk, 1968; Waltham, 2015). The size of the tide, and hence tidal friction, has varied through time because of the lunar recession. However, the main controller of the size of the tide is ocean basin geometry (Blackledge et al., 2020; Green et al., 2018). The combined effects on recession by topography and lunar distance was recently investigated in detail by Daher et al. (2021), but they limited their deep-time tidal simulations to the time slices discussed in Green et al. (2017). The results in Daher et al. (2021) supported previous work (Green et al., 2017; Williams, 2000) which strongly indicated that tidal friction, and hence the lunar recession rate, has varied through time and been much weaker than present day rates for most of Earth’s history.

A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00013-3

Copyright © 2023 Elsevier Inc. All rights reserved.

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Beyond orbital evolution, the tides, or more specifically the dissipated tidal energy, have a profound impact on the Earth System. They affect the ocean overturning circulation, which is a major redistributor of heat globally, by providing some of the energy which powers the mixing needed to support deep water formation at high latitudes ( Johnson et al., 2019; Munk, 1966; Wilmes et al., 2021). The same mixing processes support vertical fluxes of carbon and nutrients in the ocean and so play a major role in global biogeochemical cycles (Sharples et al., 2007; Tuerena et al., 2019; Williams et al., 2013). The tides have also been proposed to be drivers of key evolutionary events in Earth’s history, e.g., by regulating daylength to allow for the oxygenation of the Earth System (Klatt et al., 2021), as a driver of the environment leading to the radiation of vertebrates onto land (Balbus, 2014; Byrne et al., 2020), and as controls on sedimentation rates which leads to tidally dominated structures (Devries Klein, 1971; Yu et al., 2008). In this chapter, we will evaluate the evolution of tides over the past 540 Myr at an unprecedented temporal resolution, set by tectonic reconstructions. It is noted that similar simulations have been done (Kagan and Sundermann, 1996), but at an accuracy far below what can be achieved today, particularly in terms of tidal dissipation rates. We hypothesize that the tides have indeed remained weak throughout the Phanerozoic, except for short bursts – lasting 10 Myr or so – of tidal maxima when ocean basins are resonant (Davies et al., 2019; Green et al., 2017, 2018). We postulate that this will confirm the supertidal cycle proposed by Green et al. (2018), which is linked to the supercontinent cycle, and thus provide further evidence that the Moon is older than 1500 Myr (see, e.g., Munk, 1968, for a discussion). We will also ask if there is any possibility that the tides could have contributed to the severity of the extinctions; what were the tides up to during these key events in Earth’s history? We will do this by investigating a series of new tidal model simulations at 5 or 10 Myr intervals to explore how the tides have evolved during the eon, and we will then discuss the results from an Earth system perspective. Note that we will not discuss present-day tides, nor the very energetic tides found during the glaciations over the past 2 Myr; the latter is discussed in Chapter 9 and by previous authors (Arbic et al., 2008; Griffiths and Peltier, 2009; Wilmes and Green, 2014; Wilmes et al., 2021).

2. Tectonics The Paleozoic era started 541 Ma, after the breakup of a supercontinent, Pannotia. This led to the aggregation of the tropical continent of Laurussia,

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consisting of present-day Europe and North America, at the end of the Ordovician around 440 Ma – see Fig. 8.1A (Scotese, 2004, 2021; Scotese and Wright, 2021). Around the same time, Gondwana, consisting of present-day Africa and South America, were located over the South Pole. Laurussia was drifting south into the Rheic Ocean and closed it around 320 Ma; 10 Myr later, the supercontinent Pangea had formed, surrounded by the vast Panthalassa Ocean (Fig. 8.1E and F). Pangea was a steady feature

Fig. 8.1 Phanerozoic bathymetries. Ocean bathymetry for the 15 selected time slices, which will be discussed in more detail. Note that land is white, and the coast is marked by a thin black line. In panel (B), labels “A” and “B” refer to Laurussia and Gondwana, respectively. In panel F, the Panthalassa Ocean surrounds the supercontinent Pangea. The slices were chosen because they coincide with interesting events in Earth’s history and/or they show interesting tidal signals. Note that the colors saturate at 3000 m to allow us to see the shallow areas; the deep ocean topography is thus not seen (see Scotese and Wright, 2021 for details).

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on Earth for 130 Myr until its breakup around 180 Ma in the Mesozoic, when Gondwana, first split and Laurussia followed (Fig. 8.1G and H). As the continents started breaking apart, vast epeiric shallow shelf seas were formed (e.g., around 90 Ma) and persisted for long periods of time. Eventually, the continental configuration slowly began to resemble the planet as we recognize it today, as seen in Fig. 8.1N and O (Scotese, 2021; Seton et al., 2012).

3. “It’s life, Jim, but not as we know it” Life in the Phanerozoic started with a sudden evolution of higher life forms at the Cambrian explosion around 540 Ma (Zhuravlev and Riding, 2000). At the height of another supercontinent, Pangea, Earth hosted the most severe mass extinction in its history at 252 Ma ( Jurikova et al., 2020). In the 288 million years in between, Earth saw another two mass extinctions, at 440 Ma and 335 Ma (Raup and Sepkoski Jr, 1982). It has been shown that tides are weak during supercontinent stages (Davies et al., 2019; Green et al., 2018); this may have implications for the ocean’s ability to recirculate organic matter and potentially aggravating marine extinction events dominated by anoxic conditions and will be discussed later. The Paleozoic also saw the radiation of life onto land, the development of legs and lungs, and the dawn of vertebrates, all between approximately 430 and 395 Ma (Clack, 2012; Niedwiedzki et al., 2010). It has been suggested that tides were one of the mechanisms putting evolutionary pressure on lobe-finned fishes to move onto land (Byrne et al., 2020), and we comment on this hypothesis as a case study. The Mesozoic (252–66 Ma) saw dinosaurs and pterosaurs dominate life on land, and marine reptiles roam the oceans. The era also saw the evolution of flowering plants. As the oceans were opening up, vast epicontinental shelf seas formed (Fig. 8.1). During the Turonian (
94 Ma), a marine extinction event took place ( Jones et al., 2007), driven by anoxic conditions in these vast shallow seas. It all ended with a bang 66 Ma at the end of the Cretaceous period with another mass-extinction: the Cretaceous–Paleogene massextinction which killed off the non-avian dinosaurs and much more (Fortey, 1999). Considering the catastrophic cause of the event – a meteorite impact (Alvarez et al., 1980; Keller, 2012) – it would be far-fetched to imply any effect of tides on the severity of the event. As we move into the current era, the Cenozoic, mammals and birds filled the niches left after the

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extinction event, and Earth began to move closer to what it looks like today (Fortey, 1999).

4. The ups and downs of phanerozoic tides The tides at 420, 400, and 380 Ma were simulated by Byrne et al. (2020), 250 Maa was simulated by Green et al. (2017) and Daher et al. (2021), 120 Ma was included in Laugi
e et al. (2020), Green and Huber (2013), and Green et al. (2017) simulated 55, 25, and 2 Ma. All these simulations showed weaker tides than at present, particularly for 250 Ma, but with a suggestion of a peak before 420 Ma. This would fit the concept of the supertidal cycle (Green et al., 2018) and is one of the points we will explore here. However, it was recently suggested that the past 50 Ma have been more energetic than suggested in Green et al. (2017, 2019) based on rhythmites from 56 to 54 Ma (Zeebe and Lourens, 2019). Consequently, we will put some effort into discussing the tides over the past 55 Ma in an attempt to settle the tidal energetics argument. Because Pangea started to break up after the Great Dying of 252 Ma, we get an opportunity to discuss tidal energetics during one and a half supercontinent cycles and validate if the supertidal cycle proposed by Green et al. (2018) exists. We are aware that we simulate 250 Ma, but the extinction took place at 252 Ma. It is unlikely in a quiscent state like this that the changes in tectonic configuration over 2 Myr will have any major impact on the tides. The supertidal cycle will thus be discussed in more detail, and we will show how these variations in the tides made minor extinction events worse, allowing for the formation of fossil fuel reserves ( Jones et al., 2007).

5. Methods 5.1 Tidal modeling The simulations were made using OTIS – the Oregon State University Tidal Inversion Software – a dedicated tidal model which has been used extensively to simulate deep-time, present day, and future tides (see, e.g., Blackledge et al., 2020; Byrne et al., 2020; Davies et al., 2019; Egbert et al., 2004; a

We note that this should really be 252 Ma, but the nearest reconstruction is for 250 Ma, and it is unlikely that the changes in tectonic configuration will have changed enough over the 2 Ma to make any difference.

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Green et al., 2017; Green and Huber, 2013; Green, 2018; Wilmes and Green, 2014) and it has been benchmarked against other forward tidal models and shown to perform well (Stammer et al., 2014). OTIS provides a numerical solution to the linearized shallow water equations,
 ∂U (8.1) + f  U ¼ gHr η  ηSAL  ηEQ  F ∂t ∂η rU¼0 (8.2) ∂t in which U ¼ u H is the tidal volume transport (u is the horizontal velocity vector and H is the water depth), f is the Coriolis parameter, g is acceleration due to gravity, η is the sea-surface elevation, ηSAL is the self-attraction and loading elevation, ηEQ is the elevation of the equilibrium tide, and F the tidal energy dissipation term. The latter is a sum of two components, FB and FW. F B parameterizes bed friction and FW represents energy losses due to tidal conversion, i.e., the energy transferred into a baroclinic tide. The first term is represented using the standard quadratic law, FB]CD u ju j, where CD ¼ 0.003 is a dimensionless drag coefficient. The tidal conversion term is given by F W ¼ C U, with a conversion coefficient, C, given by (Green and Nycander, 2013; Zaron and Egbert, 2006) NHN ðrH Þ2 (8.3) 8πω Here, γ ¼ 50 represents a dimensionless scaling factor representing unresolved bathymetric roughness, NH is the buoyancy frequency at the seabed, N represents the vertical average of the buoyancy frequency, and ω is the frequency of the tide. The buoyancy frequency, N, is given by N2 ¼  g/ρ ∂ ρ/∂z, where ρ is the density. The density stratification is uncertain for most ancient oceans, so we used the values of N based on a statistical fit to observed present day values presented by Zaron and Egbert (2006). Consequently, N(x,y) ¼0.00524exp(z/1300), where z is the vertical coordinate, and the constants 0.00524 and 1300 have units of s1 and m, respectively. To evaluate potential effects of this uncertainty, some timeslices had simulations done with values of γ altered to parameterize changes in ocean stratification. C ðx, yÞ ¼ γ

5.2 Reconstructions The paleo-bathymetries came from Scotese and Wright (2021) and are described in more detail in Scotese (2021). The original data was supplied

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at 1/10° horizontal resolution, but for computational reasons they were averaged to 1/4° in both latitude and longitude (this also matches the resolution of the simulation in Chapter 7 and Green et al., 2017, 2020). All bathymetries effectively ran from 89°S to 89°N in latitude due to the introduction of land always covering the poles due to the convergence of the grid cells there. Note that outside of near-resonant states, tidal simulations are relatively insensitive to small-scale topographic changes and the blocking of the poles (Egbert et al., 2004; Wilmes and Green, 2014). The details for each era are summarized below and described in more detail in each section. We will not be able to exactly match timings of events with our reconstructions. The latter are available at 5–10 Myr resolution so events occurring in between reconstructed time slices are represented by the nearest one (e.g., 252 Ma is discussed using the 250 Ma slice). The effects of this on the tides should be minimal because major changes in tides, driven by tectonics, occur on time scales of around 5 Myr (Davies et al., 2019).

5.3 Simulations All time slices were simulated for the M2, S2, and K1 constituents, although focus is mainly on the M2 in the following because it dominates the tidal signal. Due to the uncertainty in the reconstruction, we did a suite of sensitivity simulations for every 50 Myr. We covered uncertainty in ocean stratification by doing simulations with γ in Eq. (8.3) set to 25 or 100, and we tested sensitivity to forcing by simulating the same slices, with γ ¼ 50, using present day forcing. We also did a series of simulations with halved and doubled values of γ to account for uncertainty in the stratification of the ocean for the past, but the effects of this were small and we will only discuss it for a few select time periods below. Because of tidal friction, the Earth is spinning down and, to conserve angular momentum of the Earth-Moon system, the Moon is receding (Daher et al., 2021; MacDonald, 1964). Consequently, we changed the forcing parameters when relevant for our simulations. For the Paleozoic and Mesozoic, we used forcing parameters set to those representing the beginning of each era (540 and 250 Ma, respectively) – see Table 8.1 for a summary. For the Cenozoic, from 66 Ma onwards, we used present day forcing as the changes in the forcing would have been small. Also, the results justify this (see below), because the effect of changing the forcing is actually quite small for the Phanerozoic tides. Not having the forcing change

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Table 8.1 Forcing parameters used in the model simulations. Era

Time span in simulations [Ma]

Daylength [h]

M2 period [h]

M2 forcing factor

Paleozoic Mesozoic Cenozoic

540–250 250–65 65–0

21.16 22.70 24.00

11.01 11.77 12.42

1.11 1.05 1.00

throughout the simulations for each era is a simplification, but we opted for the largest difference in forcing to present day. Consequently, for the Palaeozoic we used a daylength of 21.16 h, leading to a lunar period of 11.01 h and a tidal forcing 11% above present day whereas the Mesozoic forcing included a daylength of 22.70 h, a lunar period of 11.77 h, and a tidal forcing 5% above present day. To test the sensitivity to forcing, we also did simulations for every 50 Myr from 100 to 500 Ma and a few select time slices with major changes in tides, with PD forcing. The model output was used to compute tidal dissipation rates, D, as the difference between the time average of the work done by the tide generating force (W) and the divergence of the horizontal energy flux (P) (Egbert and Ray, 2001): D¼WrP

(8.4)

The work and flux are given by (using angular brackets to denote averages over a tidal cycle) 
 W ¼ gρ U  r ηEQ + ηSAL (8.5) and P ¼ gρhUηi

(8.6)

5.4 Present day validation The present-day control simulations from the 0 Ma bathymetry gives a rootmean-square error of 14 cm for the M2 tidal amplitudes compared to the altimetry-constrained TPXO9 model (see https://www.tpxo.net for details). A simulation with a reduced-detail (degenerate) present-day bathymetry, with a resolution similar to reconstructed bathymetries from 50 Ma, produced an error of about 20 cm (Green et al., 2017), and also hosts an M2 dissipation rate that is some 75% higher than present day simulations due to a lack of deep-ocean bathymetry. Consequently, our simulations

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using reconstructions probably overestimate the tidal dissipation rates (see Blackledge et al., 2020 for further discussion). In this chapter, we normalize dissipation rates using the 0 Ma control simulation back to 65 Ma. After that, we use the 4.5 TW from the degenerate present-day simulation in Green et al. (2017) to normalize the dissipation rates.

6. Results 6.1 Paleozoic (541–252 Ma) This is, to a large extent, a quiescent period, except for a maximum in the tidal amplitudes at 430 Ma; see Fig. 8.2 for example time slices of M2 amplitude maps and Fig. 8.3 for spatially averaged M2 tidal amplitudes. This is not too surprising as the period covers the break of one supercontinent and the formation of another, and the supercontinent stage has been shown to be tidally quiescent due to non-resonant ocean basins (Blackledge et al., 2020; Davies et al., 2019; Green et al., 2018, 2020). The reason we only see one maximum during the cycle is because Laurussia aggregated in the middle of the Panthalassa ocean whilst Gondwana became a southern hemisphere continent centered over the South Pole (Fig. 8.1). This drift pattern maintained a large open (super-)ocean – Panthalassa – which could not host large tides because its size does not allow it to be resonant for the tidal forcing at the time (Arbic et al., 2009; Green et al., 2010). As Laurussia assembled in the middle of the super-ocean around 430 Ma, relatively short-lived local resonances were set up around the continent (Fig. 8.2B and C and Byrne et al., 2020). Interestingly, the resonance in the Rheic ocean suggested by Balbus (2014) has not been identified in our simulations. A closing ocean will be resonant at some point, but it is possible that the 5 Myr temporal resolution is still too coarse to identify it or that the resonance is weak. As the assembly of Pangea continued as we approach 310 Ma, the tides became weaker and weaker due to a lack of resonances, with globally averaged amplitudes only 30–50% of present values (Fig. 8.3). The M2 tidal dissipation rates almost map onto the amplitude (Figs. 8.4 and 8.5), but with 430 Ma being the most dissipative period (Fig. 8.4). These dissipation rates are weaker than today (Fig. 8.5) because the resonances are relatively local, whereas the present-day maximum has an entire basin – the North Atlantic – near resonance (Egbert et al., 2004; Platzman, 1975). This means that at present, there can be large dissipation rates over a larger area, as opposed to the one around 450 Ma.

Fig. 8.2 Phanerozoic amplitudes. Modeled M2 tidal amplitudes for the selected time slices (see Fig. 8.1 for the associate bathymetries). Note that the color scale saturates at 2 m and that the results come from simulations with the respective tidal forcing for each era in Table 8.1.

Fig. 8.3 Averaged tidal amplitudes. The solid line shows horizontally averaged M2 tidal amplitudes for each of the simulated time slices using the relevant forcing for that era (see Table 8.1), whereas the dash-dotted line show simulations for select time slices using present-day (PD) forcing parameters. The dotted line is the average amplitude at present day from the TPXO9 database (see www.tpxo.net).

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Fig. 8.4 Phanerozoic M2 dissipation rates. As in Fig. 8.2 but showing tidal dissipation rates for the selected time slices. Note that the color scale saturates at 0.05 W m2 and how quiescent the deep ocean is in all slices.

For the remainder of the Paleozoic, the tides are very quiescent in our simulations. The globally integrated dissipation rates remain well below a third of present-day rates as Pangea assembles at 310 Ma, further supporting the idea that the supertidal cycle is linked to the supercontinent cycle (Davies et al., 2019; Green et al., 2018). When Pangea is in place, the tides are the least energetic we have seen in all our simulations, covering a total of 1750 Myr of history if we include the future 250 Ma discussed in Davies et al. (2019) and Chapter 10. Could these weak tides have aggravated the mass extinction event that took place at 252 Ma?

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Fig. 8.5 Phanerozoic dissipation rates. Shown are the horizontally integrated M2 tidal dissipation rates from the model simulations, normalized by the simulated dissipation rate (2 TW) for present day from 65 to 0 Ma, and the 4.5 TW from Green and Huber (2013) before 65 Ma (the normalization is changed due to the change in resolved topography). Note that glacial tidal dissipation rates are not included (see Chapter 9), but present day is.

During the Paleozoic, the difference between using forcing for 540 Ma and present day is relatively small, particularly for the quiescent period from 420 Ma onwards (Fig. 8.5). Interestingly, simulations with PD forcing overestimate the dissipation rates in the Cambrian, only to underestimate them during the Ordovician and Silurian, and PD forcing does not lead to a tidal maximum at 430 Ma but at 440 Ma (Fig. 8.5). Byrne et al. (2020) simulated 420–380 Ma using forcing for 400 Ma and suggested that there was a maximum before their period of investigation. We confirm that here in the simulations with 540 Ma forcing but not with PD forcing and we argue that using altered forcing more closely resembling that for the period under investigation is key. The reason for the differences between these sets of simulations lies in the periods for tidal resonances: the topographic configuration for 430 Ma allowed for local resonances when the tides were driven with a period of 11.01 h and not for a 12.42 h (present day) period.

6.2 Mesozoic (252–66 Ma) As we move into the Mesozoic, the tides remain quiet for over 100 Ma. That is, until 150 Ma when the tides suddenly – within a few million years – light up and become very large and energetic (Figs. 8.2–8.5). We see a 60 cm globally averaged M2 amplitude, which is 67% higher than at present, with tides in the Proto-Pacific basin being twice the size of today. This shows that amplitudes are not necessarily a good proxy for dissipation: the large

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amplitudes at 150 Ma occur in the deep ocean which makes them less dissipative than if they were in shallow water. This is because the tidal currents would be smaller in the deep ocean due to a larger water column to move the tide through. We argue here that the amplification at 150 Ma is due to tidal resonance in both hemispheres. Resonances occur when the tidal period is close to the period of the natural oscillation of a basin, which means that the wavelength of the tidal wave, L, or half multiples of it, matches the size of the basin (Arbic et al., 2009; Platzman et al., 1981): n L ¼ cT (8.7) 2 pffiffiffiffiffiffi where n is an integer larger than 0, c ¼ gH , the speed of the tidal wave and T is the tidal period (here T ¼ 11.77 h; note that H ¼ 5000 m in these simulations). The width of the proto-Pacific at the equator in our 150 Ma simulation is 30,000 km, which equates to 3.2 M2 wavelengths, suggesting an amplification but not a perfect resonance. However, at 20°S the basin is about 28,200 km, which is almost exactly three wavelengths in width, and at 40°N it is 1.5 wavelengths wide. This is further enhanced by the Tethys being near a half-wave length resonance: it spans approximate 5000 km, which is with 200 km of the resonant wave length for a 5000 m deep ocean. This shows that the amplification of the tide at 150 Ma is due to basin-scale resonances in the basins. The maximum at 150 Ma is short lived – not more than 10 Myr – after which the amplitudes rapidly decrease to levels below present day, but are still larger than before the maximum because the Tethys remains resonant for longer than the proto-Pacific (details of surrounding time slices not shown; Figs. 8.2 and 8.3). The latter part of the period, the Cretaceous, was home to extensive epicontinental shelf seas (Fig. 8.1). In contrast to many shelf seas on Earth today, these seas were vast and with small tidal amplitudes and associated dissipation rates in most places (Fig. 8.2D and E 8.4; see below for more details about the dissipation rates). By the Turonian extinction event at 95 Ma, the average global tidal amplitudes were around 30–35 cm (Fig. 8.3). The Turonian extinction was marine and included widespread anoxia in the epicontinental shelf seas (Kerr, 1998; Meyers et al., 2012). Our results again point to a poorly ventilated shallow ocean at the time (see the case study section below for a further discussion). Unsurprisingly, the tidal dissipation rates remain low for the first 100 Myr of the period at 11% of PD levels (Fig. 8.4 and 8.5), only to rapidly increase to around present-day levels at 150 Ma (the average dissipation for the whole

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Mesozoic is 34% of present-day rates). It may seem wrong that tidal amplitudes 60–70% larger than at present, and in a large part of the ocean, will only generate a dissipation rate equal to the rates at present day. The reason for this is that the dissipation depends on the square (for tidal conversion) or cube (for bed friction) of the tidal current, which in turn depends on the tidal amplitude but is inversely dependent on depth. This is because a certain tidal amplitude will have to transport a certain volume of water through a basin, and in a deep water-column the currents can be smaller than in a shallow water column because you have more space to transport the water through. In the Mesozoic tidal maximum at 150 Ma, the large amplitude tides were in the 5000 m deep proto-Pacific, leading to dissipation rates a fraction of those found for the same amplitudes in shallow water.

6.3 Cenozoic (66–0 Ma) During the Cenozoic, the Atlantic continued to open, the Pacific continued to close, and Australia and India continued their journeys north. These changes in the ocean basin geometries allowed the tides to gradually become more energetic as the Atlantic widened. The sensitivity simulations for 65 Ma, using present-day and Mesozoic forcing (i.e., from 250 Ma), give the same dissipation rates (Fig. 8.5). This points to the forcing being a second-order effect if the basin is not in a near-resonant state. As we move towards the greenhouse world of the Eocene, 55–45 Ma, we have tidal dissipation rates at 60–80% of present-day rates. These results agree with those in Green and Huber (2013): their 55 Ma dissipation was 45% of present day’s whereas the one presented here (Fig. 8.5) is slightly more energetic at 60% of PD (see the case study section below for a further discussion). Note that Green and Huber (2013) and Green et al. (2017) normalized their results with a different, less detailed present-day topography than the one used here, and as a result, their present-day simulation was more energetic than what is presented in this chapter. The reduced detail in their bathymetries led to an enhanced dissipation rate in the deep ocean that is more amplified in the near resonant present-day state; Blackledge et al. (2020) explores this in some detail. As we move towards the present, into the coolhouse world we are used to, we find tidal dissipation rates larger than present rates from 20 to 5 Ma, with peak rates 40% above present-day rates at 10 Ma. This maximum ended abruptly between 10 and 5 Ma. The reason for the maximum is partly because of a slightly enhanced tide in the North Atlantic (Fig. 8.4L–N), but mainly due to a resonance around Australia and

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New Zealand that breaks after 10 Ma as they continue north. In our simulations, 25 Ma sees dissipation rates equal to present day rates, whereas Green et al.’s (2017) 25 Ma simulation, using a different reconstruction, obtained dissipation rates of about half of their degenerate present-day rates, and some 80% of their realistic present day rates (note that here, we normalize with realistic present-day rates back to 65 Ma, and the degenerate rates for periods earlier than 65 Ma). We then enter the period over the last 2 Myr when Earth has gone through a series of glaciations (EPICA Community, 2004), leading to the largest tides in the 1500 Myr period we can simulate (see Chapters 6–10 in this book). This is again due to the resonant properties of the North Atlantic – a basin which is very close to a state of perfect resonance (Egbert et al., 2004; M€ uller, 2008; Platzman, 1975; Platzman et al., 1981). These very energetic tides from 20 Ma to 10–20 Myr into the future represent a very long tidal maximum, far longer than the ones identified in the past, because of the geometry of the North Atlantic as it opens. It is a very long ocean, bounded by continents on both sides for more than 11,600 km. It is also home to large shelf seas such as the European shelf, which are highly dissipative because of the large tidal currents there (Egbert and Ray, 2001; Taylor, 1920). During previous tidal maxima, the resonances were more localized (giving a smaller area for the energy to dissipate in) or in the deep ocean (which means less dissipation for the same size amplitudes; see the discussion about the 150 Ma maximum above). It appears that Earth is currently in a unique tidal state because of the unique sizes and shapes of the ocean basins. The new results here reconcile the differences between Zeebe and Lourens (2022) and Green et al. (2017): they are likely both correct. Green et al. (2017) investigated 55 Ma, 25 Ma, and 2 Ma – all of which are relatively quiescent and argued from that that the last 55 Myr has been tidally quiet. At the time, they were not aware of the rapid onset and decline of the tidal maximum around 20 Ma, which leads to an average dissipation rate of the past above present-day rates (note that the average dissipation over the past 1500 Myr in out simulations is 47% of present day).

6.4 Other constituents Focus so far has very much been on the lunar (M2) tide, simply because it is the dominant one today. And so it has been throughout the Phanerozoic: the principle solar constituent, S2, largely map onto M2 in terms of shape

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relative dissipation

3 S2 K1

2

1

0 –500

–450

–400

–350

–300

–250 –200 time [Ma]

–150

–100

–50

0

Fig. 8.6 Evolution of relative tidal dissipation rates for the S2 (solid) and K1 (dashed) constituents using present day forcing. The present day S2 and K1 dissipation rates used to normalize the data are 0.26 and 0.30 TW, respectively. The diurnal rate matches that from altimetry constrained estimates quite well, whereas S2 is underestimated by a factor of nearly 2. Note that the M2 rates we obtain are also underestimates for present-day (although with 20%), suggesting that the bathymetry used may lack detail (cf. Blackledge et al., 2020).

(cf. Figs. 8.5 and 8.6), with a large S2 maximum around 150 Ma with very energetic S2 tides compared to today (Fig. 8.6), and another maximum at 450 Ma, when M2 is at a weak peak (Fig. 8.5). This is not surprising because the two constituents are so close to each other in frequency that when an ocean basin is resonant for the lunar tide, the solar tide will be near-resonant as well. The main diurnal constituent, K1, has been larger over the past 60 Myr than over the entire Phanerozoic (dashed line in Fig. 8.6), and it was very weak during Pangea – 310-180 Ma – when the ocean surrounding Pangea was too large to host anything but near-equilibrium tides. So why is K1 large now? The answer is the Pacific. The Pacific is large enough to host relatively large diurnal tides, and because of the size of the basin (it currently covers more than 30% of Earth’s surface) a lot of energy can dissipate there, even if the energy levels are lower per area unit than the M2 tide in the Atlantic. The K1 maximum around 20 Ma coincided with the semi-diurnal maximum and is an interesting feature, we propose should be explored in the future.

7. Case studies 7.1 The Devonian The radiation of vertebrates from the ocean to land is a major evolutionary event that only happened once in Earth’s history (Clack, 2012). It has been

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suggested that one potential driver, among many, of this was the tides (Balbus, 2014): a large spring-neap range could lead to lobe-finned fishes being stranded in rock pools at spring high tide. Those with large appendages may have had an advantage because they could potentially drag themselves back to the ocean by “pool hopping” (Romer, 1933). The presence of a reasonably large spring-neap cycle – over 1 m – at the locations of early tetrapod evidence, e.g., along southern Laurussia and in eastern Gondwana (see Fig. 8.1B for locations; Laurussia is the continent just south of the Equator and Gondwana is the continent covering the South pole), was confirmed by Byrne et al. (2020) and supported by tidal proxies. Their results are redrawn in our Fig. 8.7C and D alongside the results from the simulations presented here (Fig. 8.7A and B). Our new simulations show similar magnitude signals along the south coast of Laurussia, where there are also tidal amplitudes well over 2 m in the nearshore, lending support to the idea that tides were one

Fig. 8.7 Devonian tidal ranges at 400 Ma. Shown are the resulting M2 amplitudes and S2 ranges from simulations using the reconstructions here (A–B) and redrawn results from Byrne et al. (2020) for 400 Ma (C–D). Note that the color range is different in this figure to that in Fig. 8.2, and that the S2 range shown in panels (B) and (D) is defined in terms of the difference in sea level between springs and neaps. (Credit: Data in panels C ande D redrawn from Byrne, H.M., Green, J.A.M., Balbus, S.A., Ahlberg, P.E., 2020. Tides: a key environmental driver of osteichthyan evolution and the fish-tetrapod transition?: Siluro-Devonian tides and evolution. Proc. R. Soc. A Math. Phys. Eng. Sci. 476. https://doi.org/10.1098/rspa.2020.0355.)

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process leading to evolutionary pressure that initiated the transition of vertebrates from the sea to land.

7.2 The Eocene The Eocene lasted between 56 and 33.9 Ma, and the early part of the epoch hosted an enigmatic greenhouse climate with very warm poles but a weak meridional temperature gradient (Covey and Barron, 1988; Hollis et al., 2012; Sloan et al., 1995), something coupled climate models have struggled to simulate (Huber and Caballero, 2011; Huber and Sloan, 2001; Winguth et al., 2010). The meridional heat transport would have had to increase by 300% to facilitate this temperature structure, and it was suggested that enhanced abyssal mixing could drive a stronger Eocene overturning circulation which could account for the reduced gradients (Korty et al., 2008; Lyle, 1997). Green and Huber (2013) showed that a larger fraction of the Eocene tidal energy than today dissipated in the abyssal Pacific, and that these rates were large enough to drive the mixing required to sustain the suggested overturning circulation. Because the greenhouse ocean was home to a stronger vertical stratification, Green and Huber (2013) used values of the buoyancy frequency in Eq. (8.3) that were up to 10 times larger than at present (or rather, the product of the vertical average and the buoyancy frequency at the bed in Eq. (8.3) was a factor ten larger). Consequently, we repeated our simulations for 55 Ma with γ ¼ 250 and γ ¼ 500; the resulting dissipation fields for γ ¼ 50 and γ ¼ 500 are shown in Fig. 8.8 and summarized in Table 8.2. There is indeed an enhanced abyssal dissipation rate with increased γ (i.e., stronger vertical stratification), although the globally integrated dissipation rates are relatively similar (see Table 8.2 for a summary). In the abyssal ocean, however, the simulation with present day stratification (γ ¼ 50) dissipates 0.36 TW below 500 m depth, and 0.20 TW of that is lost in the abyssal Pacific, whereas with γ ¼ 250, the numbers are 0.87 TW globally and 0.48 TW in the deep Pacific. With γ ¼ 500, as suggested by Green and Huber (2013), the rates are 1.07 TW in the global abyssal ocean and 0.58 TW in the abyssal Pacific. The results for our γ ¼ 500 simulation are more in line with Green and Huber’s (2013) abyssal rates of 1.19 TW globally and 0.56 TW in the Pacific. The differences in the global rates can be attributed to a difference in bathymetry, where the new simulations contain more detail in shelf seas, allowing for an enhanced dissipation there at the cost of dissipation in the abyssal ocean.

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Fig. 8.8 Eocene (55 Ma) M2 dissipation rates. Eocene dissipation sensitivity using γ ¼ 50 (A; present day values) and γ ¼500 (B; enhanced values) in Eq. (8.3) to represent stronger vertical stratification in the Eocene greenhouse world.

Table 8.2 Eocene parameters.

Value of γ

50 250 500 Green and Huber (2013)

Total dissipation [TW]

Abyssal (below 500 m) [TW]

Abyssal Pacific (below 500 m) [TW]

1.99 1.99 1.94 1.44

0.36 0.87 1.07 1.19

0.2 0.48 0.58 0.56

Summary of the Eocene sensitivity simulations. Note that γ ¼ 50 represents present-day conditions

The explicit effects of increased abyssal dissipation rates on the Eocene Ocean circulation and climate are yet to be investigated. It was shown that for the Turonian extinction, the mixing rates had only a small effect on the climate (Laugi
e et al., 2020). However, the Eocene Ocean may have been more sensitive to abyssal mixing because the lack of a Drake Passage made the abyssal temperature stratification much more sensitive to abyssal mixing rates. This is because there was no other mechanism to remove deep water (Toggweiler and Samuels, 1998). Also, the reduced surface temperature gradient meant that the relative contribution of the abyss to the circulation would have been greater than at present, and the differences in dissipation (compared to present) are larger in the Eocene than Turonian. The results presented by Green and Huber (2013) and seconded here stress the importance of understanding abyssal mixing rates in past climates, and the need to include them in numerical models.

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7.3 Extinctions Tides drive vertical mixing through two mechanisms. Bed friction introduces turbulent mixing upwards from the bed, which can outcompete the heat input through the sea surface and lead to vertically well mixed areas in shallow water (Simpson and Bowers, 1984; Simpson and Hunter, 1974). Under these circumstances, anoxia is unlikely to happen, and even if the water column is stratified but the currents are large, they will ensure organic material entering the lower part of the water column can be dispersed. The second mechanism is through tidal conversion, i.e., the generation of internal tidal waves (Bell, 1975; Nycander, 2005; Stigebrandt, 1999; Vic et al., 2018; Zhao et al., 2016). In shelf seas these waves predominantly exist in the thermocline, which they drive vertical fluxes across. Because the waves propagate, sometimes over large distances (Alford, 2003; Alford et al., 2007; Vic et al., 2019), they can assist in ventilating the interior of basins far from their original source. The Permian–Triassic extinction event, or “The Great Dying” occurred around 251.9 Ma ( Jurikova et al., 2020), although here it is represented by the reconstruction for 250 Ma. It is the most severe of the known extinction events, removing more than half of all biological families, 83% of genera, 81% of marine species, and 70% of terrestrial vertebrate species from Earth (Sahney and Benton, 2008; Stanley, 2016). The event was triggered because of elevated temperatures due to vast amounts of CO2 emitted during the eruption of the Siberian Traps (Kaiho et al., 2021; Ogden and Sleep, 2012). In the ocean, the increased temperature led to widespread anoxia due to a stronger vertical stratification of the upper ocean, and the higher CO2 concentrations induced ocean acidification (Knoll et al., 2007; Payne et al., 2010). Anoxic conditions are a sign of a poorly ventilated water column, e.g., by a strong vertical stratification preventing a vertical exchange of oxygen from the well-ventilated surface layer to depth whilst allowing organic matter to sink from the productive surface layer down to the bottom (Stigebrandt and Aure, 1989 present a model for the oxygen consumption under these circumstances). As mentioned above, the tides at this time were at the weakest seen over the past 1500 Myr (see Fig. 8.5). Because of the contribution of tides to vertical mixing, and therefore ventilating shelf seas, it is reasonable to assume that these low energy levels had the potential to make a bad situation worse: organic matter would have accumulated in the lower part of the water column and deteriorated there, as is happening today in poorly ventilated ocean basins such as the Baltic Sea (Carstensen

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Fig. 8.9 Turonian model results. Model simulation results for the Cenomanian– Turonian boundary at 95 Ma. (A) the bathymetry used in the simulation – note that the depth scale saturates at 1000 m to highlight the cast shelf seas. (B) simulated M2 amplitudes (color) and phases (black lines). (B) The associated M2 dissipation rates.

et al., 2014; Holtermann et al., 2020). Speculative as this is, it is an interesting point to be explored in the future. The Cenomanian-Turonian interval at 94 Ma (here represented by simulations for 95 Ma) was another greenhouse period in the Cretaceous (Laugi
e et al., 2020). It hosted a key Phanerozoic carbon cycle perturbation: the oceanic anoxic event 2 or OAE2 (Huber et al., 2018; Jenkyns et al., 2004; Kerr, 1998). OAE2 was a marine extinction event and serves as another example of where the tides, or rather the lack of them, may have been a contributor to the severity of the event. Details of the tides at 95 Ma are shown in Fig. 8.9, from which it is clear that the vast epicontinental seas present on most continents were largely void of tides. This may come as a surprise – the largest tides at present are found on the shelf seas bordering the Atlantic (e.g., Fig. 8.3O) – but the shelf seas 95 Ma were so wide that the tide dissipated before it could propagate across them. Also, we come back to tidal resonances: these basins were the wrong size to host large tides, except in the northern seas of India and Australia and along the east coast of current Asia (Fig. 8.9B). We propose here that the weak tides at the time of the event may have made it worse, but the exact effects will again be left for a future investigation.

8. Summary Together with the results from Chapter 7, we now have a suite of tidal model results covering
1/3 of Earth’s history, from 1500 Ma to present day. The globally integrated dissipation rates are summarized in Fig. 8.10 and discussed above and in Chapter 7 in some detail. The average dissipation rate over the whole period is 47% of the present-day rate when using the changed forcing and 72% of the present day dissipation rate if present-day forcing was used throughout. It is well established that the tides must have

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Fig. 8.10 1500 Myr of tidal dissipation rates. Shown are globally integrated relative tidal dissipation rates from 1500 Ma to present day. The gray block arrows mark supercontinent states (“P” is Pannotia and “P’gea” marks Pangea). As in previous figures, the solid line uses the appropriate forcing for each period whereas the dashdotted line show results using present day forcing (see Chapter 7 and Table 8.1). For the Proterozoic, stars mark simulation using a daylength of 18.7 h and plus signs mark simulation with a 21.9-h day (see Chapter 7 for details). The simulations before 250 Ma were normalized using the 4.5 TW from Green et al. (2017) degenerate present day bathymetry. Simulations for 250–0 Ma use the present-day rate (2 TW) in the 0 Ma simulation here.

been less energetic throughout Earth’s history to reconcile the age of the Moon. It is paradoxical that using present-day forcing, which is weaker, gives a larger overall dissipation rate than when using the stronger forcing for each era. The reason is that the longer tidal periods of today are resonant in larger basins, which means there is a larger area available to dissipate the energy in. In contrast, with shorter day lengths and associated tidal periods of the younger Earth, resonant ocean basins were most likely smaller. It appears that the most recent tidal maximum, that we are currently in, is prolonged and very energetic due to the continental configuration. A stronger tidal forcing does not necessarily mean a more energetic tide. The results we present are produced using an established and wellvalidated tidal model and the variations over time in the tide can be explained using fundamental dynamic principles. There is an obvious shortcoming here though: there is no easily accessible database for tidal proxies. A few papers have collated tidal proxies from the literature with some success (Byrne et al., 2020; Green et al., 2020; Wells et al., 2010; Zuchuat et al., 2022) and we note here that our results match those in these papers. We propose that this gap is closed as soon as possible, and a proof-of-concept study led by the lead author has identified over 400 papers with potential proxies from tidalites and black shales.

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We now have the simulations, and they are ready to be used in other studies of the Earth system, including investigations of orbital evolution, ocean circulation and climate, and key evolution- and extinction events and their biogeochemical cycles.

Acknowledgments The authors thank Joao Duarte for his patience editing the chapter. Simulations were done on Supercomputing Wales (funded by HEFCW) and the technical support of Aaron Owen is greatly appreciated. A special thank you to the members of the Deep time tides and Earth system dynamics (DETEST) lab group and to Phil Woodworth for comments on earlier versions of the manuscript.

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CHAPTER 9

Present day: Tides in a changing climate Sophie-Berenice Wilmesa, Sophie Warda, and Katsuto Ueharab a School of Ocean Sciences, Bangor University, Menai Bridge, United Kingdom Research Institute for Applied Mechanics, Kyushu University, Kasuga, Fukuoka, Japan

b

1. Introduction Tides affect numerous processes throughout the oceans: in the intertidal zone, tides shape ecosystem composition; in coastal areas, they transport sediments and nutrients, thus, impacting benthic ecosystems and coastal morphology, and in the deep ocean, they drive climate-regulating processes. Tides are an important driver of day-to-day sea-level variability, affecting flood and storm surge dynamics (e.g., Horsburgh and Wilson, 2007) and impacting the effects of severe storms. In shallow shelf seas, tidal dynamics determine the locations of tidal mixing fronts which separate seasonally stratified waters from permanently mixed waters (e.g., Simpson and Hunter, 1974). This separation affects nutrient cycling and primary and secondary productivity (e.g., Holligan, 1981) and modulates the storage of carbon in the shelf seas, as well as affecting its exchange with the atmosphere and deep ocean (Thomas et al., 2004). In the deep ocean, when the surface (‘barotropic’) tide interacts with rough topography, a part of the tide’s energy is converted to internal (‘baroclinic’) tides which propagate through the ocean’s interior, eventually break, and then cause mixing across density interfaces (e.g., Vic et al., 2019; MacKinnon et al., 2017). Tides supply around half of the energy (approximately 1 TW) to the ocean mixing which sustains the large-scale meridional overturning circulation (e.g., Wunsch and Ferrari, 2004; Ferrari and Wunsch, 2009). These large-scale tidal processes are important for global climate patterns and ocean biogeochemical cycles (i.e., the cycling of carbon and nutrients in the interior of the ocean). Tides also impact ice sheet dynamics: they affect ice stream flow speeds (Gudmundsson, 2007), modulate grounding line location (Tsai and Gudmundsson, 2015), and basal melt races of ice shelves (Makinson et al., 2011). Tides behave like shallow water waves (e.g., Hendershott, 1977; see Chapter 4 for details). Therefore, tidal dynamics are highly sensitive to water A Journey Through Tides https://doi.org/10.1016/B978-0-323-90851-1.00009-1

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depth and ocean basin shape changes, which can alter the resonant properties of the basin, shifting its natural resonance period toward or away from that of tidal constituents (e.g., Egbert et al., 2004; Green, 2010). Inundation of land through sea-level rise can cause the formation of new shallow ocean areas. These shallow shelf seas are highly dissipative relative to the deep ocean, and this can affect regional tidal energy balances which, for example, may lead to shifts in amphidromic systems toward the newly inundated areas (e.g., Pelling et al., 2013a). The opposite effects may be seen when sea levels decrease. Sea-level changes may also alter energy losses through bed friction, which in turn, changing tidal energy fluxes and propagation speeds (e.g., Idier et al., 2017). Interactions between resonant shelf sea areas and open ocean tides can be changed by sea-level and basin shape changes (e.g., Arbic et al., 2009). During the most recent geological period, known as the Quaternary (2.5 Ma (where ’Ma’ stands for ’million years before present’) to present), large fluctuations in climate and sea level have occurred. The aim of this chapter is to outline and explain the changes in tidal dynamics that occurred in response to these sea-level changes during the late Quaternary, and to consider the implications of these changes for different parts of the climate system. We discuss five main time periods: the Last Glacial Cycle, the Last Glacial Maximum, the Deglacial and Holocene, the present, and the future. We first give an overview of the sea-level and climate changes that occurred (or which have been projected to occur) for each time period; next, describe the modeling work we have done for this chapter; and then, for each phase discuss the changes in tidal dynamics that took place drawing on new model simulations and the scientific literature.

2. Climate and sea level through the late Quaternary The Quaternary is the last and most recent subdivision of the Cenozoic Era (66 Ma to present) and is further divided into two main epochs: the Pleistocene, which ranges from 2.5 Ma to 11.65 ka (where ‘ka’ means ‘thousand years before present’), and the Holocene (11.65 ka to present) (e.g., Cohen and Gibbard, 2019). The Quaternary is remarkable for the gradual global cooling which began during the mid-Pliocene (
3 Ma) and continued until the late Pleistocene (e.g., Shackleton, 1987; Lisiecki and Raymo (2007); Spratt and Lisiecki, 2016). Pleistocene climate was marked by repeated quasi-periodic glacial cycles during which climate fluctuated between glacial phases with a cold climate and interglacial phases with climate similar to present day. Until the mid-Pleistocene, climate was characterised by

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approximately 41 kyr (where ’kyr’ means ’thousand years’) symmetric climate cycles, whereas around 1 Ma, a transition to the asymmetric 100 kyr cycles occurred, which characterise the late Pleistocene climate (see Fig. 9.1). These 100 kyr cycles are thought to be driven by variations in orbital forcing, so-called Milankovitch cycles, which alter the amount of insolation (i.e., incident solar radiation) reaching the top of Earth’s atmosphere. Modulations of the eccentricity cycle by the orbital precession together with nonlinear feedbacks in the Earth’s climate system have been identified as drivers of these 100 kyr cycles (Raymo, 1997; Abe-Ouchi et al., 2013). During the Pleistocene, there were only relatively minor movement of continents (insignificant in comparison to those which have occurred in the hundreds of million years previously) (e.g., Merdith et al., 2017). However, due to the large variations in ice volume during this epoch, large

Fig. 9.1 Overview of climate variations through the late Pleistocene. (A) CO2 from Bereiter et al. (2015), (B) Antarctic temperature anomaly with respect to present (Jouzel et al., 2007), and (C) Global mean sea-level anomaly with respect to present from Spratt and Lisiecki (2016). Marine Isotope Stages (MIS) are marked above the timeline with (yellow ¼ interglacial phases, turquoise ¼ glacial phases, magenta ¼ interstadial phases). PIG ¼ Penultimate Interglacial, PGM ¼ Penultimate Glacial Maximum, LIG ¼ Last Interglacial (MIS 5e), LGM ¼ Last Glacial Maximum (MIS 2). Time periods with high sea levels (high-stands) are marked with ‘HS’ and phases with low sea-levels (low-stands) with ‘LS’.

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fluctuations in sea level occurred from low-stands of
130 m below to highstands of
20 m above present-day levels (e.g., Spratt and Lisiecki, 2016). Glacial cycles during the late Pleistocene (
400 ka to the beginning of the Holocene
11.65 ka, see below), followed a saw-tooth pattern with shorter interglacials followed by longer cooling phases culminating in a peak glacial phase (Broecker and Van Donk, 1970) (see Fig. 9.1C). The relatively warm time periods between glacial phases are known as ‘interglacials’ (average duration of around 10–30kyr) and have tended to occur during peaks in summer Northern Hemisphere insolation. During these interglacial time periods, climate was warm and atmospheric CO2 (carbon dioxide) concentrations were around 270–290 ppm (parts per million). Note, that we are currently in an interglacial but with extraordinary atmospheric CO2 concentrations of
420 ppm which are still increasing (Fig. 9.1A and B). Since ice sheets were similarly extensive (or even less extensive) than at present, sea levels experienced high-stands during interglacials. Global mean sea level during interglacial phases has been estimated at levels similar to present or slightly above: 5–9 m higher for the Last Interglacial (
130–115 ka, O’Leary et al., 2013) and 13–20 m higher for the MIS 11 interglacial (
424–374 ka, Roberts et al., 2012) (see Fig. 9.1C). The end of interglacials (termed glacial inception) was marked by a decrease in Northern Hemisphere insolation which led to a cooling climate resulting in the expansion of existing ice sheets expanded (and the formation of new ones), atmospheric CO2 levels decreased, and global sea levels dropped (because of water being locked into ice sheets). This climate transition led into a prolonged time-period of gradual cooling (see Fig. 9.1B), eventually culminating in a ‘glacial maximum,’ which would typically then be followed by rapid warming and a shift back to interglacial climate conditions. During the Last Glacial Maximum (26.5–19 ka), ice is thought to have covered around one third of the Earth’s surface, and global mean sea level was lower by up to 130 m (e.g., Fairbanks, 1989; Spratt and Lisiecki, 2016). Glacials had a much longer duration (70–90 kyr) than interglacial phases. During the majority of the Pleistocene, sea level was therefore between 20 and 80 m lower than at present (see e.g., Spratt and Lisiecki, 2016; see Fig. 9.1C).

2.1 The Last Glacial Cycle The Last Glacial Cycle, encompassing the last glacial phase from the end of the Last Interglacial to beginning of the Holocene (115–11.7 ka) and the

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Holocene (11.7 ka–present), was characterized by an overall cooling trend overlain with repeated shifts between stadials (cold phases) and interstadials (warmer climate phases) and rapid warming at the onset of the Holocene (see Fig. 9.1). Stadial phases with low sea-levels have been identified for Marine Isotope Stages (MIS) 5d (
105 ka), 5b (
90–95 ka), 4 (
70–60 ka) and 2 (Last Glacial Maximum;
26.5–19 ka) (Figs. 9.1C, 9.2A, and 9.3) all of which coincide with minima in Northern Hemisphere summer insolation (e.g., Lambeck et al., 2002; Cutler et al., 2003). The lowest global mean sea levels during this glacial phase occurred during the latter two stadials with low-stands of 75–85 m (Siddall et al., 2008) and 120–130 m (e.g., Clark et al., 2009) below present-day values, respectively. During these phases, the small North American ice sheets expanded into one large ice sheet, the Laurentide Ice Sheet. Ice sheets also developed over Scandinavia, the Barents Sea, and Britain (the Fennoscandian and British-Irish ice sheets) and ice extent increased over Greenland and the Cordilleran range (e.g., Gowan et al., 2021). In contrast, corresponding with peaks in Northern Hemisphere Insolation, high-stands in global mean sea level occurred during MIS 5c (
100 ka), 5a (85–80 ka, both
10 m below present according to Creveling et al., 2017) and 3 (50–35 ka) with maximum sea levels around 40 m lower than present-day (Pico et al., 2017) (see Figs. 9.1C, 9.2A, and 9.3). These time periods had a much-reduced ice sheet extent—for example, during mid-MIS 3, the ice sheets covering northern Europe largely melted, the Laurentide Ice Sheet divided into a northern and a southern ice sheet, and Hudson Bay (Canada) partly flooded. After 42.5 ka, Northern Hemisphere ice sheets began to readvance, and global sea level decreased gradually by
35 m (until ca. 30 ka) and dropped more rapidly by a further 50–60 m as full glacial conditions were reached between 22.5 and 20 ka. Through the deglacial and up to the mid-Holocene, with the melting of the large glacial ice sheets, global mean sea level rose by
120 m and temperatures and atmospheric CO2 levels increased to preindustrial levels. The large sea-level fluctuations over the Last Glacial Cycle led to considerable basin shape changes (see Figs. 9.2 and 9.3). During the sea-level high-stand around 80 ka, around 9% (compared to today’s 10%) ocean was classified as shallow shelf sea area (water depths 500 m; blue), and shelf seas (water depths < 500 m; red). Dashed lines represent an alternative Laurentide Ice Sheet extent scenario for 52.5–37.5 ka (see, Gowan et al., 2021 for details). (Bottom panel) same as middle panel but for K1. Straight lines show present-day values for total, deep, and shelf dissipation, respectively, as a reference.

2.2 The Last Glacial Maximum The Last Glacial Maximum (LGM, 26.5 –19 ka) represents the coldest period in Earth’s recent history. It coincided with a minimum summer insolation in the mid- and high northern latitudes. During the LGM, large parts of the Northern Hemisphere were covered in extensive ice sheets. The

Fig. 9.3 Modeled Last Glacial Cycle M2 tidal elevation amplitudes from 72.5 ka to present (0 ka) simulated from bathymetry reconstructions from Gowan et al. (2021) (see Section 3 for details). Gray ¼ land, white ¼ ice sheets.

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Laurentide Ice Sheet occupied the North American land mass and during peak glaciation reached as far south as 37°N (the latitude of the northern borders of the US states Arizona and New Mexico). The Greenland Ice Sheet was also more extensive than at present (e.g., Vasskog et al., 2015). Across Eurasia, the Scandinavian Ice Sheet and the British Ice Sheet covered western Europe, and the Barents Sea and northern Russia were covered by the Barents Ice Sheet. Large ice sheets formed over the Cordilleran, Patagonian, Alpine and Himalayan mountain ranges, and the Antarctic Ice Sheet is thought to have extended out to the shelf break. Global sea surface temperatures were on average 6.1 °C colder than at present (Tierney et al., 2020) and surface air temperatures around 4–6 °C colder than present-day (Annan and Hargreaves, 2013; Seltzer et al., 2021). Atmospheric CO2 concentrations reached a minimum of 190 ppm (Clark et al., 2009) (Fig. 9.1A) and the proportion of carbon lost from the atmosphere was sequestered into the deep ocean during the LGM (e.g., Khatiwala et al., 2019). Driven by the formation of large glacial ice sheets which locked up huge amounts of fresh water, global mean sea level was approximately 115–135 m lower than present during the LGM (see Fig. 9.4B and 9.5A) which substantially changed the depth and shape of the ocean basins and exposed (what we know as) continental shelf seas (see Fig. 9.4B). This reduced the total surface area of the oceans by 5% and meant that LGM continental shelf area was much reduced (
75% less than at present; Boehme et al., 2012) and a larger proportion of the oceans was classed as deep ocean (see Fig. 9.5B). Additionally, the previously flooded shelf seas now formed land bridges and extensive coastal plains.

2.3 The Last Deglacial During the most recent deglacial phase (19–11.7 ka), climate transitioned from full glacial conditions to the much warmer interglacial conditions that characterize the Holocene (11.7 ka to present) (e.g., Shakun and Carlson, 2010). During this phase, most parts of the climate system underwent huge changes at rapid rates. This time period of dramatic change was driven by changes in the seasonality, which occurred because of changes in the tilt of Earth’s axis altering the difference between summer and winter insolation. This, in turn, led to changes in greenhouse gas concentrations, temperatures, ice sheet extent, sea levels, weather patterns, ocean circulation, and vegetation cover, with feedbacks occurring between many components of the

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Fig. 9.4 (A) Modeled present-day M2 tidal elevation amplitudes, (B) sea-level difference between the LGM and present-day. Areas shaded gray indicate land extent during the LGM and the dark gray shading shows present-day land extent. Modeled LGM M2 elevation amplitudes for ICE-7G bathymetry reconstructions (see Section 3 for details) assuming (C) all ice is grounded (denoted ‘gr’) and (D) floating ice is permitted (denoted ‘fl’). (E) and (F) same as (C) and (D) but showing tidal dissipation differences between the LGM and present-day. Globally integrated dissipation (Dtot), and dissipation integrated for the deep ocean (Ddeep) and the shelf seas (Dshelf) for M2 are printed on the figure.

climate system. The Deglacial is generally subdivided into a number of different climate phases (see Fig. 9.5A): the early Deglacial is referred as the Oldest Dryas (19–14.7 ka), characterised mainly by Southern Hemisphere warming (Clark et al., 2012; Shakun et al., 2012) and initial ice sheet melting. It was followed by the warmer Bølling-Allerød interstadial (14.7–12.9 ka), which can further be divided into the Bølling, the Older Dryas (not to be confused with the Oldest Dryas, which preceded this period), and the Allerød. A large increase in Northern Hemisphere temperatures occurred at the onset of the Bølling interstadial: for the first
500 years, there was a sharp rise in temperatures, followed by a period of relatively constant temperatures (the Allerød). This was followed by a pronounced global cooling, known as the Younger Dryas (12.9–11.7 ka).

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Fig. 9.5 (A) Global mean sea-level change relative to present day. Climate periods from the LGM through the deglacial to the Holocene are indicated in the top of the panel. LGM ¼ Last Glacial Maximum, OD ¼ Oldest Dryas, BA ¼ Bølling-Allerød, YD ¼ Younger Dryas. Blue arrows represent timings of Meltwater Pulses (MWP). (B) Proportion of total ocean surface area occupied by shelf seas. Globally integrated modeled M2 tidal dissipation for the (C) whole ocean, (D) deep (>500 m water depth) ocean, and (E) the shelf seas (water depths < 500 m). (F–H) same as (C–E) but for K1 (note the different y-axis scales).

This markedly cooler phase in the middle of the deglaciation represented a temporary reversal to colder glacial conditions (predominantly in the Northern Hemisphere), a cooling which is associated with a slowdown of the Atlantic meridional overturning circulation (AMOC). During the Deglacial, massive volumes of water were reintroduced into the global oceans as major ice sheets melted, increasing global mean sea level from the LGM low-stand of
130 m to near present-day levels. Sea-level rise during the early deglacial was far from uniform in time, instead it was punctuated by phases of rapid sea-level increases during times of significant ice sheet retreat and the ‘meltwater pulses’ these generated (see Fig. 9.5A). For example, the first meltwater pulse (19.5–19 ka) caused a 5–10 m increase in global mean sea level (Carlson and Clark, 2012; Clark et al., 2004 and references therein; Clark et al., 2009). Sea levels increased more slowly

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thereafter with a rise of 8–20 m in the next 4 kyr. Between 14.7 and 14.3 ka a further meltwater pulse increased mean sea level by 14–18 m over less than 500 years (Deschamps et al., 2012) associated with the rapid warming at the onset of the Bølling. During the Allerød warm period, global mean sea-level rise was less rapid than during the Bølling (Carlson and Clark, 2012) and throughout the Younger Dryas the rate further decreased to 4 mm/year.

2.4 The Holocene The Holocene is the most recent geological epoch (11.7 ka–present day) and followed on from the Younger Dryas. During the early Holocene, global temperatures were relatively stable, in comparison to the large changes during the Deglacial. That said, the Holocene Thermal Maximum (10–6 ka) represented a period of relatively warm temperatures, and was followed by a cooling trend, which persisted until the early 20th century, after which global mean atmospheric temperatures began to rise sharply, a result of anthropogenic greenhouse gas emissions. During the early Holocene (11.7–8 ka), global mean sea level increased rapidly by 40–50 m, and rose by a further 10 m during the mid-Holocene (8–4.5 ka) (Khan et al., 2019; Barnett et al., 2020) (see Fig. 9.5A). The final deglaciation of the Laurentide Ice Sheet is approximated to ca. 6.7 ka (Ullman et al., 2016), whereas melting of the Antarctic Ice Sheet is thought to have continued through to the late Holocene (Ullman et al., 2016; Cuzzone et al., 2016). Overall, mean sea-level changes during the deglacial were dominated by ice sheet meltwater input to the oceans; however, after 7 ka, this meltwater input diminished because of the loss of the Northern Hemisphere ice sheets. Around this time the contribution of glacial isostatic adjustment (GIA) became relatively more important (Khan et al., 2015), where GIA is the visco-elastic response of the solid earth to the loading and unloading by waxing and waning ice sheet masses. Large ice sheets depress the land below the ice (near-field) and mantle material is pushed away from the ice load center to form so-called forebulges in the regions adjacent to the ice sheets (intermediate-field). As an ice sheet melts, the land below the ice sheet rebounds and relative sea-level drops in the near-field. In the intermediate-field areas, the forebulges collapse, mantle material migrates back to the ice load centers, and relative sea-level increases are greater than any eustatic sea-level rise resulting from the volume of water added to the global oceans from the melting ice sheet. Far-field regions are dominated by glacio-eustacy (i.e., the sea-level changes are driven by

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the uptake or loss of water through land-based ice sheets). These processes gave rise to highly spatially variable relative sea-level adjustments during the Holocene which are still ongoing today (e.g., Khan et al., 2015 and references therein).

2.5 Late Holocene to present day In comparison to the significant variations in sea level and the resulting changes in ocean basin shapes that occurred between the LGM and the mid-Holocene, sea-level changes in the late Holocene until the 20th century were relatively very small (Fig. 9.5A). Over the last 3 kyr prior to the preindustrial period (ca. 1850–1900 CE), global mean sea level is thought to have fluctuated by as little as 8 cm (Kopp et al., 2016). Following this relatively quiescent period, the early 20th century saw the beginning of a new and rapid change in global atmospheric temperatures (and associated climate). Driven by anthropogenic greenhouse gas emissions, global surface air temperatures increased by 1.1 °C between the late 19th and early 21st century (Gulev et al., 2021). Global mean sea level rose by 1.1–1.4 mm/year during the 20th century (Dangendorf et al., 2017, Kopp et al., 2016; Fox-Kemper et al., 2021), resulting in a global mean sea-level rise of 0.2 m between 1901 and 2018. The rate of global mean sea-level rise accelerated further still, to 3.7 mm/year for 2006 to 2018 (Fox-Kemper et al., 2021). At global scales, this recently observed sea-level rise has been driven by a combination of redistribution of water from the cryosphere and land into the ocean (predominantly from the melting of ice sheets and glaciers), thermal expansion of the oceans (because water expands as it heats up), and changes in land water storage (e.g., Church et al., 2013; Fox-Kemper et al., 2021), where all of these factors can, to a large extent, be attributed to anthropogenic pressures. Changes in sea level are far from spatially uniform across the globe due to the different drivers of the sea-level change; indeed, in some places, relative sea level are falling, despite global mean sea-level rise. For example, ongoing GIA processes from the deglacial ice sheet loss is causing relative sea-level to decrease over much of north-east North America, northern Greenland, Scandinavia, and a large proportion of Antarctica. Surrounding areas are experiencing sea-level increases whereas far-field areas show only small changes. Regionally important contributions to sea level change include sterodynamic changes which are caused by changes in ocean density and circulation driving the redistribution of mass, heat and salt in the oceans

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(Frederikse et al., 2020; Melet et al., 2018 and refs therein; Dangendorf et al., 2019; Gregory et al., 2019). Contemporary ice loss drives more short-term adjustments in sea-level by alterations in gravity, rotation, and deformation (e.g., Gregory et al., 2019; Gomez et al., 2010).

2.6 Future Present-day atmospheric CO2 concentrations are higher than at any point over the last two million years. Increases in so-called ‘greenhouse gasses’ (e.g., CO2, methane, nitrous oxide, and fluorinated gasses) since the preindustrial era have been driven predominantly by economic growth and increases in the world’s population. During this time, natural changes in radiative forcing (e.g., solar or volcanic) have been negligible relative to the anthropogenic changes, whereas atmospheric CO2 concentrations have increased by
140 ppm (up by 50%). Atmospheric warming due to anthropogenic greenhouse gas emissions is projected to increase global mean surface temperatures by between 0.2–1.0 °C (low emission scenario) and 2.4–4.8 °C (high emissions scenario) by the end of the 21st century (Lee et al., 2021). Due to these increase in atmospheric temperatures, global mean sea-level will continue to increase over the 21st century because all components driving sea-level (e.g., ocean heat content, ice sheet mass, or land water storage) are projected to contribute to increasing sea levels. Global mean sea-level is projected to rise by between 0.28 and 0.55 m (low emissions scenario) and 0.63–1.02 m (high emissions scenario) by the end of the century relative to 1995–2014 levels (Fox-Kemper et al., 2021). Relatively low uncertainty is associated with projections up to the year 2050, however, projections for beyond 2050 are much more uncertain, due to the discrepancies in the simulated emissions pathways (Fox-Kemper et al., 2021) and large uncertainties in ice sheet response to rising temperatures (Kopp et al., 2014). Ice sheet instabilities could lead to an additional
1 m global mean sea-level rise by 2100 (under the high emissions scenario). Beyond the end of the 21st century, sea level will continue to rise even if global mean surface temperatures stopped rising because of the long timescales it takes the oceans and the ice sheets to reach an equilibrium in response to historic greenhouse gas emissions (e.g., Hieronymus, 2019; Levermann et al., 2013; Meehl et al., 2005). Due to ongoing oceanic heat uptake and the resulting thermal expansion, combined with continued ice mass loss from the Greenland and Antarctic Ice Sheets, sea-level will likely remain at a high-stand for millennia (e.g., DeConto et al., 2021).

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As mentioned previously (see Section 4.3), sea-level changes are projected to be far from uniform spatially due the different processes driving the sea-level changes (e.g., alterations in gravity, rotation, and deformation, ongoing GIA and sterodynamic sea-level changes) (e.g., Gregory et al., 2019; Kopp et al., 2014; Palmer et al., 2021; Church et al., 2013).

3. Modeling the tides during the late Pleistocene and Holocene In recent years, new bathymetry datasets and sea-level reconstructions have been made developed for the late Pleistocene and Holocene epochs; these have enabled us to both update existing simulations of tides during the Last Deglacial and to carry out new simulations for the Last Glacial Cycle. In the following sections we outline how the new simulations were carried out.

3.1 Tide model Tidal simulations were conducted with the Oregon Tidal Inversion Software (OTIS) (Egbert et al., 2004; Green, 2010). OTIS has been used extensively to simulate deep-time, present day, and future tides (Egbert et al., 2004; Green et al., 2017, 2018; Wilmes et al., 2017; Wilmes and Green, 2014). OTIS has been benchmarked against other nonassimilated tidal models (i.e., forwardstepping models that are not nudged toward observations) and shown to perform well (Stammer et al., 2014). The model numerically solves the linearized shallow water equations (Egbert et al., 2004):
∂U + f  U5gHr η  ηSAL  ηEQ  F (9.1) ∂t ∂η 2r∙U ¼ 0 (9.2) ∂t Here, U ¼ uH is the tidal volume transport (u is the horizontal velocity vector and H is the water depth), f is the Coriolis parameter, g is acceleration due to gravity (9.8 ms2), η is the sea-surface elevation, ηSAL is the selfattraction and loading elevation, ηEQ is the elevation of the equilibrium tide, and F the tidal energy dissipation term. The latter has two components, so that F ¼ FB + FW. FB parameterizes bed friction and FW represents energy losses due to tidal conversion, that is, the generation of a baroclinic tide. The first term is represented using the standard quadratic law, FB ¼ CDu j uj, where CD is a dimensionless drag coefficient set to 0.003. The tidal

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conversion term is given by FW ¼ CU, with a conversion coefficient, C, given by (Green and Huber, 2013; Green and Nycander, 2013; Zaron and Egbert, 2006): NHN ðrH Þ2 (9.3) 8πω Here, γ ¼ 50 represents a dimensionless scaling factor representing unresolved bathymetric roughness, NH is the buoyancy frequency at the seabed, N represents the vertical average of the buoyancy frequency, and ω is the frequency of the tide. The buoyancy frequency, N, is given by N2 ¼  g/ρ ∂ ρ/∂z, where ρ is the sea water density (1025 kg/m3). C ðx, yÞ ¼ γ

3.2 Bathymetries and simulations Simulations of the tides during the Last Glacial Cycle (from 80 ka to present at intervals of 2.5 kyr) were conducted with the bathymetry reconstructions from Gowan et al. (2021) which are based on RTopo-2 (Schaffer et al., 2016). The numerical tidal model was run at a horizontal resolution of ¼° for the tidal constituents M2 and K1. A second set of runs from the LGM though the Deglacial to present (26/21–0 ka) was carried out at ⅛° horizontal resolution and a with time intervals of 0.5–1 kyr. Topography changes taking into account changes in sea-level and ice sheet extent with respect to present for two different ice sheet and sea-level models (ICE-5G (VM2 L90) v.1.2 (Peltier, 2004) and ICE-7G_NA (Roy and Peltier, 2018)) were interpolated to ⅛° horizontal resolution and added to the present-day RTopo-2 bathymetry. Two different scenarios were considered for ocean-ice sheet interactions: firstly, floating ice shelves were allowed to exist (‘fl’) throughout the deglacial and Holocene; secondly, all ice was assumed to have been grounded (‘gr’) through the deglacial and into the early Holocene (8 ka). Runs were performed for M2 and K1. To avoid biasing paleo-simulations by incorporating present-day boundary conditions, the northern open boundary is set to let to be land in all simulations. Glacial cycle runs were carried out with simplified self-attraction and loading (SAL) forcing with ηSAL ¼ 0.1η, and for deglacial simulations four SAL iterations were carried out from 6 to 0 ka and five iterations for all other simulations (see Egbert et al., 2004 for details). Tidal dissipation was calculated using the method detailed in Egbert and Ray (2001).

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4. Tides during the late Pleistocene, Holocene, and into the future In the following sections we discuss the impacts of the large sea-level changes during the late Pleistocene and Holocene on the tides and provide an overview of potential future changes in tidal dynamics. For the late Pleistocene and Holocene, we discuss our new simulations in conjunction with existing literature; and for the present-day and future, we present an overview of observations and modeling work from the scientific literature.

4.1 Tides during the Last Glacial Cycle and late Pleistocene 4.1.1 Semi-diurnal tidal changes In the following, we refer to the M2 tide unless specifically mentioned. Our tide model results show that the large fluctuations in sea-level and ocean basin shape changes over the past 80 kyr have driven pronounced changes in tidal dynamics (see Figs. 9.2 and 9.3), dominated by changes throughout the Atlantic. Between 80 and 75 ka, the M2 tidal elevation amplitudes (hereafter referred to simply as amplitude, where the tidal amplitude is half the tidal range) were similar to present, with small differences seen for the Weddell Sea and in the Labrador Sea where the presence of a small split Laurentide Ice Sheet led to most of Hudson Bay laying dry. Leading up to 60 ka, strong amplitude enhancements throughout the North and Central Atlantic together with large tides in the Weddell Sea developed as the Northern Hemisphere ice sheets expanded causing sea-level to drop. Correspondingly, dissipation in the deep ocean (>500 m water depths) increased by
50% between 80 and 55 ka to nearly 2 TW, to values approximately twice as large as at present. Dissipation in the shallow shelf seas (4 m the Labrador Sea). Furthermore, strong enhancements in

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amplitudes around the margins of the Arctic basin developed during the sealevel low-stand at the peak of the Last Glacial Maximum. The maximum in Atlantic semi-diurnal tides also corresponds with a maximum in open ocean and globally integrated dissipation. In contrast, shelf sea dissipation reached its minimum of 0.7 TW (less than half of present-day) at 20 ka during the peak glacial phase and rapidly increased to near present-day values by the early Holocene, while tidal dissipation in the open ocean did not decrease to present-day levels until the mid-Holocene. 4.1.2 Changes in the principle diurnal tidal constituent For the principle diurnal tidal constituent, K1, the changes in tidal dynamics are much smaller than M2 for the part of Last Glacial Cycle discussed here. Amplitude changes are generally localized and restricted to the shelf seas in the west Pacific and around northern Australia. During the transition to the Last Glacial Maximum and through to the Deglacial (32.5–15 ka), and to a lesser extent during the sea-level low-stand around 60 ka, large K1 tides develop around Antarctica, a feature also discussed in Griffiths and Peltier (2009). Enhanced tides can also be seen in the Nordic Seas. During the two sea-level low-stands, K1 dissipation shift from the shelf seas to the open ocean with open ocean dissipation nearly doubling with respect to present during the Last Glacial Maximum, whereas shelf sea dissipation decreased more than a third lower. However, overall, global K1 tidal dissipation varied less than 0.1 TW over the last 80 kyr. 4.1.3 Implications for tidal changes during the late Pleistocene Thus far, no global reconstructions of ice sheet extent and sea-level exist further back in time (i.e., beyond 80 ka) which would enable high-resolution modeling of long-term tidal dissipation changes. However, interesting inferences for long-term changes may be drawn from the modeled Last Glacial Cycle changes discussed in the previous sections. As climate, ice sheet extent and sea-level followed repeated saw-tooth shaped cycles during the Pleistocene (see Fig. 9.1), it is likely that tidal dynamics will have followed similar patterns as shown here for the Last Glacial Cycle throughout most of the Pleistocene. This would mean that during sea-level low-stands and glacial maxima, dissipation would have been strongly enhanced in the open ocean and reduced in the near-absent shelf seas (especially for the semidiurnal tidal constituents), whereas during high-stands we would find the opposite signal. Furthermore, our simulations for the Last Glacial Cycle show that open/ deep ocean (and to a lesser extent) total dissipation was enhanced for most

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of the cycle because sea-levels were lower than present for most of the period. Sea-level reconstructions for previous glacial cycles show a similar pattern (e.g., Spratt and Lisiecki, 2016) suggesting that open ocean tidal dissipation and Atlantic tidal amplitudes may have been greater than present for large parts of the Pleistocene. This may have had important implications, e.g., for ocean mixing (see Section 2.2 for a detailed discussion) or sea-level reconstructions from sea-level index points (e.g., Gehrels et al., 1995).

4.2 Tidal dynamics during the Last Glacial Maximum Tides during the LGM were first simulated by Thomas and S€ undermann (1999), with their early modeling results indicating that semi-diurnal tides were increased in the glacial Atlantic, although sources of tidal energy losses such as internal tides remained unexplored at that point in time. Egbert et al. (2004) used a far more advanced forward tidal model which included parameterisations for the energy losses to the internal tides (see e.g., Egbert and Ray, 2000, 2001) together with SAL effects and higher resolution bathymetry. They showed some surprising results which have since been confirmed by further studies using different paleobathymetries and tide models (Arbic et al., 2004; Green, 2010; Griffiths and Peltier, 2008, 2009; Uehara et al., 2006; Ward et al., 2016; Wilmes and Green, 2014) and can also be seen in our new simulations with updated paleobathymetries. 4.2.1 Tidal elevation amplitudes Semi-diurnal (M2) tides during the LGM were strongly enhanced throughout the Atlantic, with tidal amplitudes exceeding 3–4 m in parts of the Labrador Sea together with amplitudes >2 m along the European Shelf break (Fig. 9.4C and D). Other areas with greater M2 tides than at present-day included the Saragasso Sea (southeast coast of North America), the Brazil Basin and the west African coastline to the south of the Gulf of Guinea, the Indian Ocean (north of Madagascar, the Gulf of Aden, the Arabian Sea, and the North Australian Basin). Amplitudes throughout the Pacific Ocean generally remained small (0.5 mm to 200 mm) aggregates of particulate material which “rain” down from the surface ocean into the deep, dark interior (Turner, 2015; Trudnowska et al., 2021). As well as living or dead phytoplankton cells, other material such as detritus and bacteria can become bound up in marine snow (Fig. 13.3), encouraging them to sink and be exported out of the euphotic zone (Turner, 2015). Due to the range of material that can come bound up in these fast-sinking particles there is a diversity in their appearance and color (Trudnowska et al., 2021), for example from green to pale white as healthy cells are broken down. Packed with carbon and the other nutritious organic material from phytoplankton cells, marine snow provides a feast for bacteria and protists, who rapidly colonize them. Organic material, with its high-water content, is roughly the density

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A journey through tides

Fig. 13.3 Example of a marine snow particle. Various detrital material makes up the particle, including dead phytoplankton cells and broken off parts of larger plankton. Black scale bar represents roughly 100 μm (0.1 mm).

of seawater and so material that adds density, such as opal and calcium carbonate, act as ballast to increase the sinking speed of marine snow and shorten its transit time from the upper ocean to the interior and sea-bed (Klaas and Archer, 2002). This means that aggregates formed from diatoms and/or coccolithophores, with their opal shells and calcium carbonate scales, sink faster than those simply composed of organic material. While phytoplankton represent the primary producers of the ocean, fixing carbon using energy from the sun, the herbivorous grazers of the ocean are called zooplankton and come in two general size categories, the small (